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This book can be ordered in a paper bound reprint from:
The Educational Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212 USA Toll Free: 1-866-880-5373 E-mail: [email protected]
Peer Reviewer: Prof. Wenpeng Zhang, Department of Mathematics, Northwest University, Xian, Shaanxi, P. R. China. Prof. Xingsen Li, Ningbo Institute of Technology, Zhejiang University, Ningbo, P. R. China. Prof. Chunyan Yang and Weihua Li, Guangdong University of Technology, Institute of Extenics and Innovative Methods, Guangzhou, P. R. China. Prof. Qiaoxing Li, Lanzhou University, Lanzhou, P. R. China. Ph. D. Xiaomei Li, Dept. of Computer Science, Guangdong University of Technology, P. R. China. Copyright 2012 by The Educational Publisher, translators, editors and authors for their papers Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 9781599731902 Words: 175,000 Standard Address Number: 297-5092 Printed in the United States of America
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8
8
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
1 {0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 SmarandacheSmarandache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 'uSmarandache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 aSmarandacheEulerI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 aSmarandacheEulerII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 'uSmarandache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 'uSmarandacheSmarandacheI . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.1 9yG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.2 'K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 'uSmarandacheSmarandacheII . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 'uSmarandache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Smarandache9Smarandache' . . . . . . . . . . . . . . . . 27
3.1 aSmarandache' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 'uSmarandachepSmarandache . . . . . . . . . . . . . . . . . . . . . 29
3.3 'uSmarandache9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 'uSmarandachepSmarandache . . . . . . . . . . . . . . . . . . 39
3.5 'uSmarandacheV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 a2Smarandache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7 Smarandache91aSmarandache . . . . . . . . . . . . . 48
3.8 'uSmarandache LCMSmarandache . . . . . . . . . . . . . . . . . . . . . . 51
3.9 'uSmarandache LCM9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 SmarandacheS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Smarandache LCM 'SI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Smarandache LCM 'SII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Smarandache 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 Smarandache1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.2 Smarandache V1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Smarandache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Smarandache 3n iS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
I
-
8
4.6 Smarandache kn iS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.7 Smarandache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1 kFibonacciLucas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Dirichlet L-n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 2Dirichlet L- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.1 'un5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.2 'un5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
II
-
c
c
q, 5, AO5|. /
, X|u, {3u, y
\N|, 3I, [email protected]|. F1
-
c
IV
-
1 {0
1 {0
35kK, vk)6, {7Z;[F.Smarandache
J108)K, ;[,. 0
k'SmarandacheSmarandache9'K#?.
?nSmarandacheS(n)mn|m!, =
S(n) = min{m : m N, n|m!}.
lS(n)
-
Smarandache9'K
2
-
1 SmarandacheSmarandache
1 SmarandacheSmarandache
2.1 'uSmarandache
?n, F. SmarandacheS(n)mn|m!.~XS(n)cAS(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) =
3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 10, S(11) = 11, S(12) = 4, . 'uS(n){5, z[1, 7], p2E.
'uS(n)5, N?1, k(J, ~XF.
Luca3z[2]?
A(x) =1
x
n6x
S(n)
e.OK, A(x)r.O. 3z[3]y?
n>1
n
S(1) S(2) S(n)
, Je:
: ?x > 1, kC
LS(x)
n6x
ln S(n) = ln x ln ln x + O(1).
'u, 8qvk
-
Smarandache9'K
?k > 1, k
LS(s) 1x
n6x
ln S(n) = ln x + C + O
(1
lnk x
)
,
CO~.
yn2.1, IeAn.
nnn 2.1 ?p, kC
p6x
ln p
p= ln x + O(1),
p6x
ln p = x + O( x
ln x
)
.
y:z[4 6].e|^dnny. kLS(x).O.
, dS(n)5, ?nkS(n) 6 n, dEuler
(z[6]n3. 1), k
LS(x)=1
x
n6x
ln S(n) 61
x
n6x
ln n
=1
x
n6x
x
1
ln ndy 61
x
x
1
ln ydy +ln(x + 1)
x
=ln x 1 + 1x
+ln(x + 1)
x.
=
LS(x) =1
x
n6x
ln S(n) 6 ln x + O(1). (2-1)
g, |^n5OS(x) 1x
n6x
LnS(n) e.. ?
n > 1, w,nkfp, n = p n1, udS(n)5
S(n) > p
ln S(n) = ln S(p n1) > ln p.
ld(z[6]n3.17), k
LS(x)
=1
x
n6x
ln S(n)
4
-
1 SmarandacheSmarandache
>1
x
pn16x
ln p
=1
x
p6
x
ln p
n6xp
1 +
n6
x
1
p6 xn
ln p
p6
x
ln p
n6
x
1
=1
x
p6
x
ln p (
x
p+ O(1)
)
+
n6
x
1 (x
n+ O
( x
n ln x
))
p6
x
ln p (
x + O(1))
=1
x
x
p6
x
ln p
p+ O
n6
x
ln p
+ x
n6
x
1
n+ O(x)
x
p6
x
ln p + O(
x)
=1
x
{
x ln
x + O(
x) + x ln
x + O(x)
x
(x + O
(x
ln x
))}
=ln
x + ln
x + O(1)
= ln x + O(1).
=
LS(x) > ln x + O(1). (2-2)
d(2-1)9(2-2)k
LS(x) = ln x + O(1).
un2.1y.
2.2 aSmarandacheEulerI
u?n, SmarandacheS(n)mn|m!,=S(n) = min{m : m N, n|m!}.Smarandache59A^+
-
Smarandache9'K
nnn 2.2 Euler5, =u?pmn, Kk(mn) =
(m)(n).
nnn 2.3 n = p11 p22 pkk nIO), Kk
(n) =k
i=1
pi1i (pi 1).
nnn 2.4 n > 2, K7k2|(n).
yyy: (1)enkf, p, Kp > 22|p 1.qdn2.3(p 1)|(n),2|(n).
(2)envkf, n > 2, 7kn = 2k,ku1.dEuler
5(n) = 2k1(2 1) = 2k1, k2|(n). d(1)(2), n2.4.
nnn 2.5 n = p11 p22 pkk nIO), K
S(n) = max{S(p11 ), S(p22 ), , S(pkk )}.
yyy: z[7].
nnn 2.6 upk, kS(pk) 6 kp.AO/, k < p, kS(pk) = kp.
yyy: z[7].
nnn 2.7 y = p2(p 1) p, p, K > 3, yN4O.
yyy: y = (p 1)p2 ln p p, > 3, y > 0,(.en2.2y.rk = 7\(n) = S(nk),=
(n) = S(n7). (2-4)
w,n = 1(2-4)). e?n > 16.
n > 1n = p11 p22 pkk nIO),Kdn2.5k
S(n7) = max{S(p711 ), S(p722 ), , S(p7kk )} = S(p7r). (2-5)
dn2.2,
(n) = (pr)
(n
pr
)
= pr1(p 1)(
n
pr
)
. (2-6)
(2-4))(2-6),
pr1(p 1)(
n
pr
)
= S(p7r). (2-7)
6
-
1 SmarandacheSmarandache
r = 1, ep = 2,d(2-7)
S(27) = 8 = (n
2
)
,
=n = 32,
(32) = 25 24 = 16 6= S(327),
n = 32(2-4)).
ep = 3,K
S(37) = 18 = 2(n
3
)
,
n2.4g, (2-4)).
n, ep = 5, 7, 11, (2-4)).
ep > 11,K
S(p7) = 7p = (p 1)(
n
p
)
,
2|p 1, n2.4g, (2-4)).r = 2, ep = 2, d(2-7)
S(214) = 16 = 2(n
4
)
,
=(n4 ) = 8, Kn = 64,
S(647) = S(242) = 46 6= 32 = 26 25 = (64),
n = 64(2-4)).
ep = 3, K
S(314) = 30 = 6(n
9
)
,
n2.4 g, (2-4)).
ny,ep = 5, 7, 11, 13,(2-4)).
ep > 17,K
S(p14) = 14p = p(p 1)(
n
p2
)
,
=k
7 =p 1
2
(n
p2
)
,
n2.4g, (2-4)).
r = 3, ep = 2,
S(221) = 24 = 22(n
8
)
,
7
-
Smarandache9'K
Kn8 = 9, n = 72,
(72) = (23 22)(32 3) = 24, S(727) = S(221 314) = 30 6= 24 = (72),
n = 72(2-4)).
ep = 3,K
S(321) = 45 = 18( n
27
)
,
n2.4g, (2-4)).
ep = 5, 7, 11, 13, 17, 19,ny(2-4)).
ep > 23,dn2.6k,
S(p21) = 21p = p2(p 1)(
n
p3
)
,
=
21 = p(p 1)(
n
p3
)
,
p > 23g,(2-4)).
r = 4, ep = 2,
S(228) = 32 = 23( n
16
)
,
=(
n16
)= 4, n = 80,
(80) = (24 23)(5 1) = 32, S(807) = S(228 57) = 32 = (80),
n = 80(2-4)).
ep = 3,
S(328) = 60 = 33 2( n
81
)
,
U, d(2-4)).
ep > 5,dn2.6k,
28p > S(p28) = p3(p 1)(
n
p4
)
,
=
28 > p2(p 1)(
n
p4
)
,
p > 5g, (2-4)).
r = 5,ep = 2,
S(235) = 42 = 24( n
32
)
,
8
-
1 SmarandacheSmarandache
n2.4g, (2-4)).
ep > 3,
35p > S(p35) = p4(p 1)(
n
p5
)
,
=
35 > p3(p 1)(
n
p5
)
,
p > 3g, (2-4)).
r > 6,ep = 2,
S(27r) = 2r1( n
2r
)
,
=2r1|S(p7r), dSmarandachen2.5!2.6 , U, (2-4)).
ep > 3,K
7rp > S(p7r) = pr1(p 1)(
n
pr
)
,
=
7r > pr2(p 1)(
n
pr
)
> pr2(p 1),
dn2.7,U, (2-4)).
n: k = 7, (n) = S(nk)=k)n = 1, 80.dn2.2y
.
2.3 aSmarandacheEulerII
un, SmarandacheS(n), (n)Euler, !UY
SmarandacheEuler
(n) = S(nk)
). 3!, (2-3)3k = 7), !3d:
(2-3)3k > 8)L9)K. =kXeAn.
nnn 2.3 k = 8, (2-3)=k)n = 1, 125, 250, 289, 578.
nnn 2.4 k = 9, (2-3)=k)n = 1, 361, 722.
nnn 2.5 k > 10, (2-3)=3k).
nnn 2.6 en = p11 p22 pss nIO),
S(pkr) = max{S(pk11 ), S(pk22 ), , S(pkss )},
9
-
Smarandache9'K
Kp > 2k + 1k > 1, r > 2, (2-3)); 2k + 1, (2-3) =k2
), n = 2p2.
e5ny. kn2.3y. rk = 8\(2-3),
(n) = S(n8). (2-8)
w,n = 1(2-8)).e?n > 16.
n > 1n = p11 p22 pss nIO), Kdn2.6k
S(n8) = max{S(p811 ), S(p822 ), , S(p8kk )} = S(p8r) (2-9)
dn2.3,
(n) = (pr)
(n
pr
)
= pr1(p 1)(
n
pr
)
. (2-10)
(2-8)-(2-10),
pr1(p 1)(
n
pr
)
= S(p8r). (2-11)
p = 2, r = 1,d(2-11)
(n
2
)
= S(28) = 10, (2-12)
d(2-12)(
n2
)> 1. Kn7fq,
S(q8)
=
18, q = 3
35, q = 5
49, q = 7
8q, q > 7
(2-13)
(2-9)(2-12)g. p = 2, r = 1, (2-8)).
(32) = 25 24 = 16 6= S(327), n = 32(2-8)).p = 2, r = 2, d(2-11)2(n
4) = S(216) = 18, =(n
4) = 9, n2.4g,
p = 2, r = 2, (2-8)).
ny,p = 2, r = 3, 4, 5, 6, 7, 8, (2-8)).
p = 2, r > 8, dn2.6k
8r >1
2S(28r) =
1
22r1
( n
2r
)
= 2r2( n
2r
)
> 2r2 =1
42r,
=32r > 2r, g. p = 2, r > 8, (2-8)).
p = 3, r = 1, d(5)22(
n3
)= S
(38)
= 18, =(
n3
)= 9, n2.4g
,p = 3, r = 1, (2-8)).
10
-
1 SmarandacheSmarandache
ny,p = 3, r = 2, 3, 4,(2-8)).
p = 3, r > 5, dn2.6k
8r >1
3S(28r) =
2
33r1
( n
3r
)
= 2 3r2( n
3r
)
> 2 3r2 > 2er2 >
2
(
1 + (r 2) + 12(r 2)2 + 1
6(r 2)3 + 1
24(r 2)4 + 1
120(r 2)5
)
> 8r,
g.p = 3, r > 5,(2-8)).
p = 5, r = 1, d(2-11)4(
n5
)= S(58) = 35,g,(2-8)).
p = 5, r = 2,20(
n25
)= S(516) = 70, =2( n
25) = 7, g, (2-8)).
p = 5, r = 3,100( n125) = S(524) = 100, =
(n
125
)= 1, n = 125, 250,
(125) = (53) = 100 = S(524) = S(1258),
(250) = (2 53) = 100 = S(524) = S(2508),
n = 125, 250(2-8)).
p = 5, r = 4, 53 4(
n625
)> S(532) = 130,(2-8)). n
p = 5, r > 4, (2-8)).
p = 7, r = 1,d(2-11)6(
n7
)= S(78) = 49,g,(2-8)).
np = 7, r = 2, (2-8)).p = 7, r = 3,
72 6( n
243
)
> S(724) = 147,
(2-8));p = 7, r > 3, (2-8)); p = 11, 13, r > 1, (2-8)).
p = 17, r = 1,d(2-11)16(
n17
)= S
(178)
= 17 8,g,(2-8)).
p = 17, r = 2,k17 16(
n172
)= S
(1716
)= 17 16,=
(n
289
)= 1,n = 289, 578,
(289) = (172)
= 272 = S(1716
)= S(2898),
(578) = 272 = S(1716
)= S
(5788
),
n = 289, 578(2-8)).
p = 17, r > 3, du17r1 16(
n17r
)> S
(178r
),(2-8)).
p = 19, r = 1,k18(
n19
)= S
(198)
= 8 19,g,(2-8)).n,p = 19, r = 2, 3,(2-8));p = 19, r > 4, 19r > 19r218( n19r ) >
18 19r2,g. (2-8)).p > 19, r = 1,k(p 1)(np ) = S(p8) = 8 p,(p, p 1) = 1, e
11
-
Smarandache9'K
k 8pp1 = 2k(k N+), =p =k(p1)
4 , g.(2-8)).
p > 19, r > 2,dn2.6k
8r > pr2(p 1)(
n
pr
)
> 22 23r2,
w,g.(2-8)).
n:k = 8,(2-8)=k)n = 1, 125, 250, 289, 578.y
n2.3.
n,^q{yn2.4.
eyn2.5. ?k,n > 1, n = p11 p22 pss
nIO), dS(n)
S(nk) = max{S(pk11 ), S(pk22 ), , S(pkss )} = S(pkr).
q
(n) = (pr)
(n
pr
)
= pr1(p 1)(
n
pr
)
, (2-14)
d(2-3)
pr1(p 1)(
n
pr
)
= S(pkr). (2-15)
p 6 S(pkr) 6 krp.
epr1(p 1)( npr
) = krp,Kd(n)95pr1(p 1)|krp,=pr2(p 1)|kr.-y = pr2(p 1) kr, p,yr.K
y = (p 1)pr2 ln p k.
p > 2k + 1r > 2,y > 0,dyN4O,y > 0.
(2-15)p > 2k+1r > 2).==3knU(2-13)
.
, dn2.2-2.6,(n2.4!n2.5y{n2.6.
2.4 'uSmarandache
SmarandachedF.Smarandache3 Only Problems Not Solutions ?, S(n) = min{m : n|m!}.5
-
1 SmarandacheSmarandache
d?1L.
Erdos[15]QuAknkS(n) = P (n),P (n)n
f. u8c(JAleksandar Ivic[25], y
N(x) = x exp
{
2 log x log log x
(
1 + O
(log log log x
log log x
))}
,
pN(x)LkLxvS(n) = P (n).Mark
FarrisPatrick Mitchell[26]SmarandacheS(n)3e.O,
Xe(J:
(p 1) 6 S(p) 6 (p 1)( + 1 + logp ) + 1;
Jozsef Sandor[27]Smarandache)5, y
S(m1 + m2 + + mk) < S(m1) + S(m2) + + S(mk)
S(m1 + m2 + + mk) > S(m1) + S(m2) + + S(mk)
3). Lu Yaming[28]?y3e3|).
'uS(n)Z(n)kN5, AOZ(n)du5K5
5k(J.+3[29]JeK:kk
n|d1
S(d); Z(n)Majumdar3[30]Jeo
:
(1)Z(n) = 2n 1=n = 2k,kK;(2)Z(n) = n 1=n = pk,pu3;(3)XJnULn = 2k/, KkZ(n) 6 n 1;(4)?nkZ(n) 6= Z(n + 1).!K), u+JK),
MajumdarJo(. `(5, keA
n.
nnn 2.8 eS(n) = P (n),nIO)n = p11 p22 prr Pr+1(n), @o7kr+1 =
1;u1 6 k 6 r,7kk 6 P (n) 2.
yyy: er+1 > 1, KkP2(n)|n,n|S(n)! = P (n)!,KP 2(n)|P (n)!.P (n)g
.kr+1 = 1.
n!fp,=vp|n!, p+1 n!,Kk =
l=1
[npl
]
(y
13
-
Smarandache9'K
[31]). dukpkk |P (n)!,dk
k 6
l=1
[P (n)
plk
]
6
l=1
P (n)
plk=
P (n)
pk 1,
dpk 6= 2w,kk 6 P (n) 2; pk = 2, Kkt2t < P (n) < 2t+1,dk
k 6t
l=1
[P (n)
2l
]
6
t
l=1
P (n)
2l= P (n) P (n) 2
2t+1< P (n) 1,
kdk 6 P (n) 2.
\BZ(n), #Z(n) = min{m : n|m(m + 1)}, w,
NnZ(n) = Z(n),nZ(n) = Z(2n), d
'.e'uZ(n)5.
nnn 2.9 n = pk, Z(n) = n1;nkf,K7kZ(n) 6n2 1.
yyy:dZ(n)Z(n) 6 n 1, n = pkkpk|Z(n)(Z(n) + 1), k
pk|Z(n)
pk|Z(n) + 1,
dZ(n) > pk 1 = n 1,dkZ(n) = n 1.nkf, dn7)pu1
, n = mk,(m, k) = 1,dk, K733b, b
ukK{Xmb l mod kmb l mod k, d7kZ(n) 6min{mb, mb 1}.5k|(mb + mb),dkk|(b + b),kb + b = k,duk, 7kmin{b, b} 6 (k1)2 ,d
Z(n) 6 mk 1
2=
n
2 m
26
n
2 1.
dneAn.
nnn 2.7 -
f(n) =
d|n
1
S(d),
@o3n 6 xN(x), f(n),
N(x) = x exp
{
2 log x log log x
(
1 + O
(log log log x
log log x
))}
,
14
-
1 SmarandacheSmarandache
=f(n)3Ak.
yyy:dAleksandar Ivic[25](J, IyS(n) = P (n), f(n).
eS(n) = P (n),dn2.8, nIO)n = p11 p22 prr P (n), k 6
P (n) 2,nPn = mP (n),dk
f(n)=
d|n
1
S(d)=
d|m
1
S(d)+
d|m
1
S(dP (n))
=
d|m
1
S(d)+
d|m
1
P (n)
=
d|m
1
S(d)+
rk=1(k + 1)
P (n).
I5ud|m, S(d) < S(n) = P (n), k + 1 6 P (n) 1 < P (n), du1, , 1fuP (n),u1ff
uP (n),d, d\U.n2.7y.
nnn 2.8 (i)Z(n) = 2n 1=n = 2k, kK;(ii) Z(n) = n 1=n = pk, pu2;(iii)XJnULn = 2k/, KkZ(n) 6 n 1.
yyy:(i)n = 2k(=k, k = 0w,), dkZ(n) = Z(2n),d
n2.9Z(n) = 2n 1; eZ(n) = 2n 1,7kn, KZ(n) = Z(n) 6 n 1, bn 6= 2k,2nkf,dn2.9Z(n) = Z(2n) 6 2n
21 = n1,
g, n = 2k.
(ii)n = pk, pu2, dkZ(n) = Z(n), dn2.9Z(n) =
n 1; eZ(n) = n 1,w,nU, bn 6= pk,Knkf, dn2.7Z(n) = Z(n) 6 n2 1,g, kn = pk, pu2.
(iii)enULn = 2k/, K7kn, dkZ(n) = Z(n) 6 n 1,Knkf, dn2.9dk
Z(n) = Z(2n) 62n
2 1 = n 1.
nnn 2.9 ?nkZ(n) 6= Z(n + 1).
yyy:bknZ(n) = Z(n + 1) = m,K3k, knk = (n + 1)k = m(m+1)2
,d
u(n, n+1) = 1,dkn+1|k, n|k,dkn+1 6 k, n 6 k,m(m+1)2 > n(n+1),km > n + 1,dn2.8K7kn = 2t, n + 1 = 2t
,Un = 1, dw,
kZ(1) 6= Z(2).
15
-
Smarandache9'K
nnn 2.10 ?x > 1, k
x2J
j=1
aj
logjx+ O
[x2
logJ+1x
]
6
n6x
Z(n) 63
8x2 + O
[x2
lnx
]
,
aj =(j+1)
2.
yyy: kym:dn2.8,n = 2k, Z(n) = 2n1,n = pk,Z(n) = pk 1, nkf, Z(n) 6 n 1; nk2fkZ(n) 6 (n1)2 , d
n6x
Z(n)=
k6[x2]
Z(2k) +
k6[x2]
Z(2k + 1) + O(x)
6
k6[x2]
2k 1 +
k6[x2]
k +
p6x
p 12
+
k6ln x
2k + O(x)
=3
k6[x2]
k +
p6x
p 12
+
k6ln x
2k + O(x)
=3
8x2 + O
(x2
ln x
)
.
eyn: P (n)|nZ(n)(Z(n)+1)
2,kP (n)|Z(n)(Z(n) + 1),
Z(n) > P (n) 1,d:
n6x
Z(n)>
n6x
(P (n) 1)
=
n6x
P (n) + O(x)
=x2J
j==1
aj
logjx+ O
(x2
logJ+1x
)
.
z[25].
2.5 'uSmarandacheSmarandacheI
2.5.1 9yG
?nF. SmarandacheS(n)mn|m!=S(n) = min{m : m N, n|m!}. lS(n)
-
1 SmarandacheSmarandache
5,N?1,k(J[7,27,28,33,36]. ~X, Lu Yaming
aS(n))5[28] , yTk|), =y?
k > 2
S(m1 + m2 + + mk) = S(m1) + S(m2) + + S(mk)
k|)(m1, m2, ,mk).Jozsef Sandor[27]?`?k > 2, 3|
(m1, m2, , mk)
v
S(m1 + m2 + + mk) > S(m1) + S(m2) + + S(mk).
, q3|(m1 + m2 + + mk)v
S(m1 + m2 + + mk) < S(m1) + S(m2) + + S(mk).
d, M[36]k'S(n)(J[36]. =yC
n6x
(S(n) P (n))2 = 2(32)x
32
3 ln x+ O
(
x32
ln2 x
)
,
P (n)Lnf, (s)LRiemann Zeta-.
y3,Z(n)
Z(n) = min
{
m : m N, n
m(m + 1)
2
}
.
TkSmarandache. 'u5, ,8,
k
-
Smarandache9'K
nnn 2.12 ?n,
Z(n) + 1 = S(n)
=n = p m, p, mp12 ?. =m|p12 .
w,, Tn.)K(A)9(B). =yk
), z)N/. , AO3m[1, 100], Z(n) =
S(n)k9),On = 1, 6, 14, 15, 22, 28, 33, 66, 91.uK(B),w,Z(n)+
1 = S(n)3m[1, 50]k19 ), O
n = 3, 5, 7, 10, 11, 13, 17, 19, 21, 23, 26, 29, 31, 34, 37, 39, 41, 43, 47.
|^{ny.
kyn2.10. , n = 1, Z(n) = S(n). n = 2, 3, 4, 5,
w,vZ(n) = S(n). u, bn > 6vZ(n) = S(n), Z(n) =
S(n) = k. dZ(n)9S(n), k, nve
:
n
k(k + 1)
2, n|k!. (2-16)
k,y3(2-16)k+1U. XJk+1,k+1 =
p,u3n|p(p1)2(n, p) = 1, n|p1
2.ln
p2
i=1
i =(p 1)(p 2)
2.
k = p 1n|k(k+1)2 g(n, p) > 1, dup, p|n. 2dun|k!, p|k!. U, p = k + 1,pU(p 1)!, ly3(2-16)k + 1U.
g, y3(2-16)kk. kk+12
, ek, Kk), k = a b, a > 1, b >1a 6= b. u5(k, k+1
2) = 1,Jk = a b|(k 1)!k+1
2|(k 1)! 2dun
k(k+1)
2 n|(k 1)!. kn|k!g. k,, k = p.duk, p > 3, lp, 2p, , p1uk 1z(k 1)!, udnk(k+1)2 E,n|(k 1)!. kg, k.
(kkk = p,dnp(p+1)2
p!, n
p+12 ,w,kS(n) < p;n = pZ(n) 6= S(n). n = p m, mp+12
18
-
1 SmarandacheSmarandache
?u1.
y3yn = p m, mp+12?u1, kZ(n) = S(n).
dw,kS(pm) = S(p) = p.mp1
i=1
i = p(p1)2
, Kmp+12g!
Z(pm) = p, lZ(pm) = S(pm).
, y3kZ(n) = S(n) = k. ^y{5y(. b
3k = 2m, Z(n) = S(n) = k = 2m,KdZ(n)9S(n)
,nk(k+1)2 = m(2m + 1) 9(2m)!. dc2m + 1U, K
(n, 2m + 1) = 1, n2m1
i=1
i = m(2m 1), w,2mn
m(2m + 1)g!(n, 2m + 1) > 1,d5p = 2m + 1|n, l2dn|(2m)!p = 2m + 1|(2m)!, g!2m + 1U, ymU, KNn|(2m 1)!, 2mn|(2m)!g!lmp, k = 2p. un|p(2p + 1)! 9n|(2p)!. , nup(2p + 1)?U!`?k|p(2p + 1), UkS(k) = 2p. u, n2.10y.
y3yn2.11. n2.10y{q, pVL. b
nvZ(n) + 1 = S(n), Z(n) + 1 = S(n) = k. udZ(n)9S(n)
Jk,
n
k(k 1)2
, n|k!. (2-17)
w,, f(2-17)k!KJn|(k 1)!, kn|k!g. dk = p. 2dn|p(p1)2 , 5S(
p12 ) < p,
n = p m,mp12?. Nyn = p m,mp1
2?, n
vZ(n) + 1 = S(n).
(2-17)k = 2m, k 1 = 2m 1, l3nZ(n)+1 = S(n) = 2m. ,Z(n)+1 = S(n)=n = p m,mp12 ?. u, n2.11y.
2.5.2 'K
'uSmarandacheS(n)9SmarandacheZ(n)5,
?, E,3K. Buk,?1?, p0
S(n)9Z(n) k'kK.
KKK2.1 ?n,H(n)Lm[1, n]kS(n)
19
-
Smarandache9'K
. H(n)C5. :
limn
H(n)
n= 1.
KKK2.2 kkn,
d|n
1
S(d)
,
d|nLnk. ?=n = 1, 8.
KKK2.3 S(n)P (n)9
P (nS(n))5,
n6x
S(n)
P (n),
n6x
P (n)
S(n)
C, P (n)Lnf. xu, 1C
uc x,c,u1~;1ATx. ,K2.3K2.1'. XJK2.1(, @o1C.
KKK2.4 Z(n)5K, knXn = m(m+1)2 , kZ(n) = m 7.n = p11 p
22 pkk n IO
), ddF.Smarandache5
S(n) = max16i6r
{S(pii )} S(p) = u p,
p,pi, ,i, u 6 .
y35p|n9S(n) = u p,n = p n1,nv(2-19)k
Z(n) + u p = p n1. (2-20)
ky3(2-20) = 1. Kb > 2, ud(2-20)p|Z(n) =m.dZ(n) = m n = p n1m(m+1)2 , (m, m + 1) = 1,p|m.ld(2-20)p|S(n) = u p, =p1|u,lp1 6 u.,, 5S(n) =S(p) = u p, dF.SmarandacheS(n)5u 6 ,p1 6 u 6 . dpw,. XJp = 2, K > 3, p1 6 u 6 . u
kU:u = a = 2. 5n > 59S(n) = 4,dk
U:n = 12,n = 12(2)). XJnv(2-19),K(2-
20)7kS(n) = p, = u = 1.3e, -Z(n) = m = p v, K(2-20)
v + 1 = n1,
n1 = v + 1,=n = p (v + 1), Z(n) = p v. 2dZ(n)n = p (v + 1)Z(n)(Z(n) + 1)
2=
pv (pv + 1)2
,
(v + 1)
Z(n)(Z(n) + 1)
2=
v (pv + 1)2
,
5(v + 1, v) = 1,vdv + 1|pv + p p + 1,=v +1|p 1v + 1|p12 . w,
p12 ?u1fr, n = p r (2-19)).
dkZ(p r) = p (r 1).v, d(v + 1)
Z(n)(Z(n) + 1)
2=
v (pv + 1)2
,
22
-
1 SmarandacheSmarandache
(v + 1)
pv + 1
2=
(p 1)(v + 1) + v p + 22
.
dd
p 1 = (2k + 1) (v + 1).
up 1 = 2 h,h, Kv+12uhf. Ny?
r|hr < h, n = p 2 r (2-19)). dk
Z(p 2 r) = p (2 r 1).
, 5r|h. kNp 2 rm(m+1)2 .udZ(n)
Z(p 2 r) = p (2 r 1).
uyn2.13.
y3yn2.14. d5k = 2, n = 1, kZ(1) + S(1) = 2=n = 1
(2-19)). XJ(2-19)k)n > 2,Kdn2.13y{
Jn = p u, p > 5, S(u) < p. \(2-19)
Z(p u) + S(p u) = 2p u.
ddpZ(p u).Z(p u) = p v, Kv = 2u 1.dZ(n)p u
p(2u1)(p(2u1)+1)2 . lu
p12 . d, u
p12 ?u1, Z(p
u) = p (u 1), dn = p u(2-19)); up12 ?k
Z(p u) = p (2u 1),
d
Z(p u) + S(p u) = 2p u.
un2.14y.
dn2.13J. n2.13 UFermat,
pFermat , p 1vku1f.
2.7 'uSmarandache
?n, F.SmarandacheS(n)mn | m!.
23
-
Smarandache9'K
=S(n) = min{m : m N, n|m!}. {7Z;[F.Smarandache35Only Problems, Not Solutions6\,
-
1 SmarandacheSmarandache
2.2 ?n, k4
limn
PS(n)
n= 1.
!|^{ny. kOn PS(n).. n > 1, n = p11 p
22 prr LnIO), @odS(n)95
S(n) = S (pii ) = m pi. ei = 1, @om = 1S(n) = pi. ei > 1, @om > 1, KS(n). n PS(n)m[1, n]kS(n) = 19S(n)n! w,S(n) = 1=n = 1. u-M = ln n, Kk
n PS(n) = 1 +
k6n
S(k)=S(p), >2
1 6 1 +
S(k)6M
1 +
kp6n
p>M, >2
1. (2-21)
y3OO(2-21), w,k
kp6n
p>M, >2
1 6
kp26n
2p>M
1 +
kp6n
p>M, >3
1 6
M2
M, >3
k6 np
1
M2
M, >3
n
p n
ln n+
p6
n
p>M, >p
n
p+
p6
n
p>M, 36
M
n
p+
p6
n
p>
M, >3
n
p
nln n
+n
2
M1+
n
M n
ln n. (2-22)
u(2-21),, I#O{. ?p 6 M , -(p) =[
Mp1
]
, =(p)LL Mp1. u =
p6M
p(p). ?vS(k) 6
Mk, S(k) = S(p), KdS(k)kp|M !, l 6
j=1
[M
pj
]
6
M
p 1. kvS(k) 6 Mku, kLu, =d(u). k
S(k)6M
16
d|u1 =
p6M
(1 + (p))
=
p6M
(
1 +
[M
p 1
])
=exp
p6M
ln
(
1 +
[M
p 1
])
, (2-23)
25
-
Smarandache9'K
exp(y) = ey.
dn/(z[4]9[6])
(M) =
p6M
1 =M
ln M+ O
(M
ln2 M
)
,
p6M
ln p = M + O
(M
ln M
)
:
p6M
ln
(
1 +
[M
p 1
])
6
p6M
ln
(
1 +M
p 1
)
=
p6M
[
ln (p 1 + M) ln p ln(
1 1p
)]
6(M) ln(2M)
p6M
ln p +
p6M
1
p
=M ln(2M)
ln M M + O
(M
ln M
)
= O
(M
ln M
)
. (2-24)
5M = ln n, d(2-23)9(2-24)O:
S(k)6M
1 exp(
c ln nln ln n
)
, (2-25)
c~.
5 exp(
cln nln lnn
) n
lnn, u((2-21), (2-22)9(2-25)O:
n PS(n) = 1 +
k6n
S(k)=S(p), >2
1 = O( n
ln n
)
.
PS(n) = n + O( n
ln n
)
.
uny.
26
-
1n Smarandache9Smarandache'
1n Smarandache9Smarandache'
3.1 aSmarandache'
cSmarandacheS(n), mn|m!, =S(n) =min{m : m N, n|m!}. lS(n) N: XJn = p11 p22 pkk LnIO), @o
S(n) = max{S(p11 ), S(p22 ), , S(pkk )}.
'uS(n)5, z[7, 36, 53 55, 57]. 3z[58], Sandor\
SmarandacheS(n)S(n), Xe, u?n, S(n)
mm!|n, =k
S(n) = max{m : m N, m!|n}.
'uS(n)5k?1L, XJ. ~X3z[59]
Zk'S(n)
d|nSL(d) =
d|nS(d)
)5k(, =e
A =
n :
d|nSL(d) =
d|nS(d), n N
,
Ku?s, Drichlet?f(s) =
n>1,nA1ns3s 6 1u, 3s > 1,
k
f(s) = (s)
(
1 112s
)
,
SL(n)Smarandache LCM,
SL(n) = max{k : [1, 2, , k]|n, k N},
(s)LRiemann zeta.
3z[27], Sandor\SmarandacheZ(n)Xe: u?n,
27
-
Smarandache9'K
Z(n)m, n|m(m+1)2 , =
Z(n) = min
{
m : m N, n
m(m + 1)
2
}
.
lZ(n)OZ(n)cA: Z(1) = 1, Z(2) = 3, Z(3) = 2, Z(4) =
7, Z(5) = 4, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 8, Z(10) = 1, . 'uZ(n)5, N?1, k(J[4,6,6368], Z(n){
5:
a)u?9p, Z(p) = p 1;b)u?,Z(2) = 2+1 1.!0'uSmarandacheSmarandache
Z(n) + S(n) 1 = kn, k > 1 (3-1)
)5. K8cvk 1v(3-1); k > 3, (3-1)).
|^9|{ny. {, S(n) =
m. k = 1, w,n = 1v(3-1). n = 1(3-1)).
ebn > 1v(3-1). dSmarandacheZ(n)
Z(n)(Z(n) 1) + mZ(n) = nZ(n).
dd(Z(n)nmZ(n), 5m!|n, mZ(n) = qn, Z(n) = qn
m. d 1vZ(n) = n. m = 2, d(3-1)Z(n) = n 1n > 1, dn = p, > 1, p, un = p, > 1(3-1)). m = 3,
28
-
1n Smarandache9Smarandache'
=S(n) = m = 3, Kn = 6, yn = 6v(3-1).k = 1, nv
(3-1)=n = 1. k = 2, w,n = 1v(3-1).
n, bn > 1v(3-1). Z(n)dk
Z(n)(Z(n) 1) + mZ(n) = 2nZ(n).
nmZ(n), 5m!|n, mZ(n) = qn, Z(n) = qnm . d\(3-1) q
nm +m1 = 2n. dS(n) = mm!|n,ln = m!n2,
dz
k(m 1)! n2 + m 1 = 2m! n2. (3-3)
3(3-3)kw,(m 1)!, d5(3-3)1nm 1U(m 1)!. w,(m 1)!m 1=m = 1, 2, 3.m = 1, d(3-3)k = 2, dkZ(n) = 2n, dZ(n)95,
vkn > 1 vZ(n) = 2n. m = 2,d(3-1)Z(n) = 2n 1n > 1,,n = 2, > 1(3-1)). m = 3,=S(n) = m = 3,Kn = 6, yn = 6
v(3-1). n, k = 2, nv(3-1)=n = 2, > 1.
k > 3, dyn, vkn > 1vZ(n) =
kn,Z(n) = kn 1, n = 6v(3-1). (3-1)).n, Bn3.1y.
3.2 'uSmarandachepSmarandache
SmarandacheS(n)mn|m!,=S(n) = min{m :n|m!}. SmarandacheZ(n)v
m
k=1
kUn
m,=Z(n) = min{m : n|(m(m + 1))/2}.~X,Z(n)cAZ(1) = 1, Z(2) =3, Z(3) = 2, Z(4) = 7, Z(5) = 5, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 9, Z(10) =
4, Z(11) = 10, Z(12) = 8, Z(13) = 12, Z(14) = 7, Z(15) = 5, Z(16) = 3, Z(17) =
16, Z(18) = 8, Z(19) = 18, Z(20) = 15, .'uS(n)Z(n), N5,
(
J[7,29,6972]. 32.5!
Z(n) = S(n), Z(n) + 1 = S(n)
)5, ).
29
-
Smarandache9'K
z[75]?SmarandachepSc(n),
max{m : y|n!, 1 < y 6 m, (m + 1)n!}.
~X, Sc(n)cASc(1) = 1, Sc(2) = 2, Sc(3) = 3, Sc(4) = 4, Sc(5) = 6, Sc(5) =
5, Sc(7) = 10, Sc(8) = 10, Sc(9) = 10, Sc(10) = 10, Sc(11) = 12, Sc(13) = 16, Sc(14) =
16, Sc(15) = 16, . z[72] Sc(n)Z(n)m'X
Z(n) + Sc(n) = 2n,
(J, Je.
: n N+, Sc(n) + Z(n) = 2n
=n = 1, 3, p2+1, 3 + 2uu2, p > 5
,p2+1 + 2?.
!y, =en.
nnn 3.2 n,
Sc(n) + Z(n) = 2n
)Un = 1, 3, p2+1, 3 + 2uu2, p > 5
, p2+1 + 2?.
nnn 3.3 n,:
(1)n2, nSc(n) + Z(n) = 2n);
(2)n, nk3f, nSc(n) + Z(n) =
2n).
yn3.2n3.3, keAn.
nnn 3.1 eSc(n) = x N+,n 6= 3,Kx + 1un.
nnn 3.2 (1)e N+,KZ(2) = 2+1 1;(2)epu2, N+, KZ(p) = p 1.
nnn 3.3 n N+,n > 59, KkSc(n) < 3n/2.
nnn 3.4 n, 3nn + n7/12m7k.
n3.1-3.4yz[30, 75, 77].
eyn3.2. n, 65?.
30
-
1n Smarandache9Smarandache'
(1)n = 1v.
(2)n = 3, Z(3) = 2, Sc(3) = 3,3).
(3)n = 3( > 2),3 + 2, K
Sc(3) = 3 + 1, Z(3) = 3 1,
3).
(4)n = 3( > 2),3 + 2, Kdn3.1n3.2, k
Sc(3) > 3 + 1, Z(3) = 3 1,
3).
(5)n = p , > 1, p > 5, K
Z(p) = p 1
n 6= 3, 2:en = p2 , > 1, Kp 1(mod3),u
p2 1(mod3),
l
p2 + 2 0(mod3),
Kp2 + 2 U, dn3.1, dn).
en = p2+1,p > 5, p2+1 + 2 ?, d
n3.1,
Sc(p2+1) = p2+1 + 1.
dn3.2, Z(p2+1) = p2+1 1,KSc(p2+1) + Z(p2+1) = p2+1 + 1 + p2+1 1 =2p2+1,n = p2+1. ep2+1 + 2, Kdn3.1, Sc(p2+1) >
p2+1, n = p2+1).
(6)2n, n = p11 n1, (p1, n1) = 1, 1 > 1, p1 > 3 , {
n1x 1(modp11 )
k), ?{
n21x2 1(modp11 )
k), )y,K1 6 y 6 p11 1,qp11 yc{), K
31
-
Smarandache9'K
1 6 y 6(p
11 1)
2. d
n21y2 (modp11 ),
K
p11 |(n1y 1)(n1y + 1),
(n1y 1, n1y + 1)|2, u(p11 1)|(n1y 1)(p11 1)|(n1y + 1).e(p11 1)|(n1y 1),Kn = p11 n2
(n1y1)(n1y)
2 , l
Z(n) = m 6 n1y 1 6(p11 1)n1
2 1 6 n
2,
e(p11 1)|(n1y + 1), Kn = p11 n1(n1y1)(n1y)
2, l
Z(n) = m 6 n1y 6(p11 1)n1
26
n
2.
dn3.3, n > 59, Kk
Sc(n) + Z(n) 1,
Z(2) = 2 1, Sc(2) > 1,
n = 2v.
(2)n = 2kp, > 1, (2k, p) = 1, p > 3, {
4kx 1(modp)
k),{
16k2x2 1(modp)
k), )y,K1 6 y 6 p 1,qp yc{), K1 6 y 6 (p
1)2 .d
16k2y2 1(modp),
Kp|(4ky 1)(4ky + 1),(4ky 1, 4ky + 1) = 1,up|(4ky 1)p|(4ky + 1).
32
-
1n Smarandache9Smarandache'
ep|(4ky 1), Kn = 2kp4ky(4ky1)
2, lk
Z(n) = m 6 4ky 1 6 4k(p 1)2
1 6 n 2k 1 6(
1 1p
)
n.
ep|(4ky + 1), Kn = 2kp4ky(4ky+1)
2 , lk
Z(n) = m 6 4ky 64k(p 1)
26 n 2k 6
(
1 1p
)
n.
(3)n = (2k + 1)2, > 1, k > 1, K{
(2k + 1)x 1(mod2+1)
(2k + 1)x 1(mod2+1)
7k), ), {
(2k + 1)x 1(mod2+1)
), e1 6 6 2 1, K=, K
2 + 1 6 6 2+1 1,
K
2+1 6 2+1 2 1 = 2 1,
2+1 v{(2k + 1)x 1(mod2+1),
{7kv
1 6 6 2 1
), K2+1|[(2k + 1) + 1]2+1|[(2k + 1) 1].e2+1|[(2k + 1) + 1],K2+1(2k + 1)|[(2k + 1) + 1](2k + 1), l
Z(n) 6 (2k + 1) 6 (2 1)(2k + 1) 6(
1 12
)
n.
2+1|[(2k + 1) 1], nkZ(n) 6(1 12
)n.
o, u(2),(3), =
n = 2p11 p22 pkk ( > 1, i > 1, k > 2)
33
-
Smarandache9'K
IO), -
q = min{2, p11 , p22 , , pkk },
K
n = 2p11 p22 pkk > q3 ,
lq < 3
n,K
Z(n) 6 n
(
1 1q
)
< n
(
1 13
n
)
= n n 23 ,
, n, dn3.4,
Sc(n) + Z(n) < n + n712 + n n 23 < 2n,
=Tnv. n, n3.3.
3.3 'uSmarandache9
?nSmarandacheZ(n)vm
k=1
kUn
m,=Z(n) = min{m : m N+, n|m(m+1)2 }. dZ(n)NZ(1) =1, Z(2) = 3, Z(3) = 2, Z(4) = 7, Z(5) = 4, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) =
8, Z(10) = 4, Z(11) = 10, .NZ(n) 5,
(J[30,79,80]:
(a)u?n,kZ(n) > 1;
(b)u?pk, kZ(pk) = pk 1, Z(2k) = 2k+1 1;(c)en?, KZ(n) = max{Z(m) : m|n}.z[27]\Z(n)Z(n), vnU
mk=1 k
m, =Z(n) = max{
m : m N+, m(m+1)2 |n}
. ~XZ(n)cA
Z(1) = 1, Z(2) = 1, Z(3) = 2, Z(4) = 1, Z(5) = 1, Z(6) = 3, Z(7) = 1, Z(8) =
1, Z(9) = 2, . dz[27, 82], Z(n)kXe5:(d)p, qevp = 2q 1, KZ(pq) = p; ep = 2q + 1, KZ(pq) = p 1;(e)en = 3st(s?,t), KZ(n) > 2;
(f)ka, b, kZ(ab) > max{Z(a), Z(b)}.+aZ(n)9Z(n)
Z(n) + Z(n) = n (3-4)
)5, Je:
: (A)(3-4)kk). Nk)n = 6;
34
-
1n Smarandache9Smarandache'
(B)(3-4)k)7p(p > 5).
'udK, z[83]?1, y(B), ((A)Em
K. !?0?1, |^{|{y
(5, e(:
nnn 3.4 (3-4)k)n = 6;(3-4)k)n = pk,p,
k?.
yyy:?.
1)n, e45y:
1.1)n = 2k(k > 1), kZ(2k) = 2k+1 1, Z(2k) = 1.lZ(2k) + Z(2k) 6=2k,2k(3-4)).
1.2)n = 2p(p),ep = 3,KZ(6) = 3, Z(6) = 3,lZ(6) + Z(6) =
6,n = 6(3-4));ep = 5,KZ(10) = 4, Z(10) = 4,lZ(10) + Z(10) = 8 6=10, n = 10(3-4));ep > 5,KZ(2p) = 1,=Z(2p) = 2p 1, (3-4)k). KI2p|2p(2p1)2 ,dd2p 1, g.
1.3)n = 2kp(k,),e2kpm3/XA = 2B 1'X, KU
px = 2 2y 1, (3-5)
U32y = 2 px 1.px,yOu, k. , o3xy(3-5). v(3-5)xyOPXe?
b,da,c:
1.3.1)epx = 2 2y 1, KZ(n) = max{px} = pb = 2 2a 1.-
Z(n) = m = n Z(n) = 2kp pb = 2kp 2 2a + 1,
Kk
2kp
(2kp pb)(2kp 2 2a + 2)2
(3-6)
k = 1, Ka = 1,pb = 3,u8(n = 2p/;k > 1, d(3-6)
2kpb|(2kpb 1)(2k1p 2a + 1)
w,, b = b < ,U2, d(3-4)).
1.3.2)epx = 2 2y + 1,K
Z(n) = max{px 1} = pd 1 = 2 2c.
35
-
Smarandache9'K
-
Z(n) = m = n Z(n) = 2kp pd + 1 = 2kp 2 2c,
K
2kp
(2kp 2 2c)(2kp pd + 2)2
(3-7)
k = c,d(3-7)k2p|(p2)(2kppd+2),U,(3-4)d);k > c,k2kcp|(2kc1p1)(2kppd+2),p|(2kc1p1)(2kppd+2)U, ld(3-4)).
1.3.3) e3'X(3-5). KZ(n) = 1. Z(n) 6= n 1. K1.2)?, 2g.
1.4)n = 2kuv, (2k, u) = (u, v) = (v, 2k) = 1.Px,yOu,vf,z
uk. nfme3/XA = 2B 1'X, Ux =2 2z 1x = 2y 2z 1,U32z = 2x12zy = 2x1. w,3x,y,zve?9:
1.4.1)ex = 2 2z 1,KZ(n) = max{x} = x1 = 2 2z1 1.-
Z(n) = m = n Z(n) = 2kuv x1 = 2kuv 2 2z1 + 1.
5, u = ex1, Kk
2kuv
(2kuv x1)(2kuv 2 2z1 + 2)2
(3-8)
k = 1, kz1 = 1, 2ev|(2ev 1)(uv 1), ,v|(2ev x1)(uv 1)U,g; k > 1, k
2kev|(2kev 1)(2k1uv 2z1 + 1),
lg.
1.4.2)ex = 2 2z + 1, KZ(n) = max{x 1} = x2 1 = 2 2z2. -
Z(n) = m = n Z(n) = 2kuv x2 + 1 = 2kuv 2 2z2,
K
2kuv
(2kuv 2 2z2)(2kuv 2 2z2 + 1)2
, (3-9)
k = z2,k2uv|(uv2)(2kuv2k+1+1),U,g;k > z2,k
2kz2+1uv(2kuv 2)(2kuv x2 + 2),
36
-
1n Smarandache9Smarandache'
u|(2kuv 2)(2kuv x2 + 2)U, g.1.4.3)ex = 2y 2z 1,KZ(n) = max{x} = x3 = 2y3 2z3 1. -
Z(n) = m = n Z(n) = 2kuv x3 = 2kuv 2y3 2z3 + 1.
5, u = tx3, Kk
2kuv
(2kuv x3)(2kuv 2y3 2z3 + 2)2
, (3-10)
k = 1, kz3 = 1, 2tv|(2tv 1)(uv 2y3 + 1), ,v|(2tv 1)(uv 2y3 + 1)U, g; k > 1, k2ktv|(2ktv 1)(2k1uv 2z3y3 + 1), g.
1.4.4)e= 2y 2z + 1, KZ(n) = max{x 1} = x4 1 = 2y4 2z4. -
Z(n) = m = n Z(n) = 2kuv x4 + 1 = 2kuv 2y4 2z4 ,
5, v = sy4, Kk
2kuv
(2kuv 2y4 2z4)(2kuv 2y4 2z4 + 1)2
, (3-11)
k = z4, k2uv|(uv 2y4)(2kuv 2z4+1y4 + 1), U, g;k > z4,k2
kz4+1su|(2kz4su2)(2kuvx4 +2),u|(2kz4su2)(2kuvx4 +2)U, g.
1.4.5)e34'X, KZ(n) = 1.Z(n) 6= n 1,Kn 1g.
n, (3-4))k=kn = 6.
2)n, e3?:
2.1)w,n = 1(3-4)).
2.2)n = pk(p, k), ep = 3,KZ(n) = n 1, Z(n) = 2,l
Z(n) + Z(n) 6= n; ep > 5,KZ(n) = n 1, Z(n) = 1.n = pk(3-4)).
2.3)nkf, n = uv,p(u, v) = 1. x,yOu,vf.
5, u > v. eA?:
2.3.1)e3'Xx = 2y 1. Z(n) = max{x} = x1 = 2y1 1. Pu = bx1. -
Z(n) = m = n Z(n) = uv x1 = uv 2y1 + 1,
Kk
uv
(uv x1)(uv 2y1 + 2)2
,
37
-
Smarandache9'K
dd
bv
(bv 1)(uv 2y1 + 2)2
,
v
(bv 1)(uv 2y1 + 2)2
,
w,, g.
2.3.2)e3'Xx = 2y + 1. Z(n) = max{x 1} = x2 1 = 2y2. Pn = dy2.-
Z(n) = m = n Z(n) = uv x2 + 1 = uv 2y2,
Kk
uv
(uv x2 + 2)(uv 2y2)2
,
2du|(du 2)(uv x2 + 2), u|(du 2)(uv x2 + 2), w,, g.2.3.3)e3/Xx = 2y 1'X, KZ(n) = 1. -Z(n) = m = n 1, w
,nn(n1)
2 . eydmvZ(n).
d{uX 1(modv)k). u2X2 1(modv)k). )Y , K1 6 Y 6 v 1. Kv|(uY 1)(uY + 1).
v|(uY 1), kn = uv
uY (uY 1)2
,
Z(n) = m 6 uY 1 6 u(v 1) 1 < uv 1;
v|(uY + 1), kn = uv
uY (uY + 1)
2,
Z(n) = m 6 uY 6 u(v 1) < uv 1.
m = n 1vZ(n).nA, n(3-4))=n = pk, p > 5, k
.
n, n3.4.
38
-
1n Smarandache9Smarandache'
3.4 'uSmarandachepSmarandache
?n,SmarandacheZ(n)nm
k=1
km,
=Z(n) = min{
m : nm(m+1)
2
}
.~X, TcA:Z(1) = 1, Z(2) = 3, Z(3) =
2, Z(4) = 7, Z(5) = 5, Z(6) = 3, Z(7) = 6, Z(8) = 15, Z(9) = 9, Z(l0) = 4, Z(11) =
l0, Z(12) = 8, Z(13) = 12, Z(14) = 7, Z(15) = 5, Z(16) = 31, Z(17) = 16, Z(18) =
8, Z(l9) = 8, Z(20) = 15, .'u, N5,
(J,
z[85 89]. ~X, +3z[87]Z(n) = S(n), Z(n) + 1 = S(n))5, ), 'u, 32.5!k[.
3z[75], ?SmarandachepSc(n), Sc(n)vy|n!1 6y 6 mm,=Sc(n) = max{m : y|n!, 1 6 y 6 m, m + 1n!}.~X,Sc(n)cA:Sc(1) = 1, Sc(2) = 2, Sc(3) = 3, Sc(4) = 4, Sc(5) =
6, Sc(6) = 6, Sc(7) = 10, Sc(8) = 10, Sc(9) = 10, Sc(l0) = 10, Sc(11) = 12, Sc(12) =
12, Sc(13) = 16, Sc(14) = 16, Sc(15) = 16, .z[75]Sc(n)5, ye(: eSc(n) = x, n 6= 3,
Kx + 1un.
3z[91]?SmarandacheZ(n), Z(n)vm
k=1
kn
m, =Z(n) = max{
m : m(m+1)2 |n}
. z[272]Z(n)5,
(J. z[86]nm'XZ(n) + Z(n) = n
(3.3!)Sc(n) = Z(n) + n,
(J, J
):
:Sc(n) = Z(n) + n)p, p,2, p + 2. !8K, e:
nnn 3.5 Sc(n) = Z(n) + n)p, p, 2, p + 2, 9v^(2 1)n( > 1), n + 2 , n.
3ynckeAn.
nnn 3.5 eSc(n) = x Z,n 6= 3,Kx + 1un.
yyy:z[75].
dd, Sc(n)3n = 1, n = 3, 3{e.
nnn 3.6
Z(p) =
2, p 6= 31, p = 3
39
-
Smarandache9'K
nnn 3.7 en 0(mod(2 1)),KkZ(n) > 2 > 1.
nnn 3.8
Z(n) 6
8n + 1 1
2.
n3.6-n3.8yz[27].
nnn 3.9 n = p00 p11 p
22 pkk (p0 = 2, pi > 3, k > 1, i > 1)nIO),k
Z(n) 6 n nmin{p00 , p11 , p22 , , pkk }
yyy:aqu3.3!yn3.3{. n = p00 p11 p
22 pkk (p0 = 2, pi > 3, k >
1, i > 1)IO), 5y.
(i)n = 2kp, > 1, (2k, p) = 1, p > 3, d{
4kx 1(modp)
k), {
16k2x2 1(modp)
k), )y,K1 6 y 6 p1,qpyc{), K
1 6 y 6 p1
2.d
16k2y2 1(modp),
Kp |(4ky 1)(4ky + 1),(4ky 1, 4ky + 1) = 1,up |4ky 1 p |4ky + 1 .ep |4ky 1, Kn = 2kp
4ky(4ky1)
2, l
Z(n)=m 6 4ky 1
64k(p 1)
2 1 6 n 2k 1 6
(
1 1p
)
n
6n nmin{p00 , p11 , p22 , , pkk }
.
ep |4ky + 1, Kn = 2kp4ky(4ky+1)
2, lk
Z(n) = m 6 4ky 64k(p 1)
26 n 2k =
(
1 1p
)
n 6 n nmin{p00 , p11 , p22 , , pkk }
.
(ii)n = 2(2k + 1), ( > 1, k > 1),K{
(2k + 1)x 1(mod2+1)
(2k + 1) 1(mod2+1)
40
-
1n Smarandache9Smarandache'
k), ), {
(2k + 1)x 1(mod2+1)
), e1 6 6 2 1,K=, K
2 + 1 6 6 2+1 1,
K
2+1 6 2+1 2 1 = 2 1,
2+1 v{(2k + 1)x 1(mod2+1),
{7kv1 6 6 21),K2+1 |(2k + 1) + 12+1 |(2k + 1) 1,e2+1 |(2k + 1) + 1,K
2+1(2k + 1) |[(2k + 1) + 1](2k + 1),
l
Z(n) 6 (2k + 1) 6 (2 1)(2k + 1) 6(
1 12
)
n 6 n nmin{p00 , p11 , p22 , , pkk }
.
2+1|(2k + 1) 1, nk
Z(n) 6
(
1 12
)
n 6 n nmin{p00 , p11 , p22 , , pkk }
.
n(i),(ii)k, n = p00 p11 p
22 pkk (p0 = 2, pi > 3, k > 1, i > 1)IO)
, K
Z(n) 6 n nmin{p00 , p11 , p22 , , pkk }
.
eny, 5y.
(1)n = 1, Z(1) = 1, Sc(1) = 1,K1).
(2)n = 3( > 1),dn3.6, Z(3) = 2,en = 3),KSc(3) =
2 + 3,3|3 + 2 + 1,l3 + 2 + 1Un3.5g, n = 3).
(3)n = p( > 1, p > 5),dn3.6,Z(p) = 1,en = p),KSc(p) =
1 + p,p > 5, 3|p2+2,dn3.5, U, p + 2(2) ,n = p( > 1, p > 5)v.
(4)n = 2( > 1),em(m+1)2 |2,(m, m+1) = 1,Km = 1,Z(2) = 1,en = 2),KSc(2) = 1 + 2,2|(2 + 1 + 1),n3.5g,n = 2( > 1)
41
-
Smarandache9'K
).
(5)n = p11 p22 pkk (k > 2, i > 1) IO). q5y.
(i)2n,K2pii ,l2|Sc(n),env,K7L2Z(n).yZ(n),e3( > 1),(2 1)|n,KZ(n) > 2 1,e3( > 1),(2 + 1)|n,KZ(n) > 2,l
Z(n) = max{max{2k : k(2k + 1)|n}, max{2k 1 : k(2k 1)|n}}.
2n5?
1,eZ(n) = 2 1 > 1,K(2 1)|n,k|n, |[n + (2 1) + 1]. env, KSc(n) = 2 1 + n.Sc(n) + 1, n3.5g.1, eZ(n) = 2 > 1,K(2 + 1)|n,k|n, (2 + 1)|n.env,
KSc(n) = 2 + n.Sc(n) + 1, n3.5g.
, eZ(n) = 1, d > 1, K(2 1)n, len + 2,dn3.5, n ). en + 2, dn3.5, n). =(2 1)n, n + 2 n).
(ii)2|n,env, K7LZ(n),Z(n) > 2,
Z(n) = m > 2,m(m + 1)
2|n ,
K(m + 1)|n,?(m + 1)|(n + m + 1), Sc(n) = n + m + 1,n3.5g.d, Sc(n) = Z(n) + n)p, p, 2, p + 2, 9v^(2 1)n( > 1), n + 2 , n. n3.5y.
3.5 'uSmarandacheV
?n,SmarandacheZ(n)m
n|m(m+1)2 , =Z(n) = min{
m : m N, n|m(m+1)2}
.dZ
;[Smarandache3z[7]?. 'uZ(n)5, N?1,
k(J[28,36,9799]. ~X:
(1)u?n,Z(n) < n.
(2)?p > 3, Z(p) = p 1.(3)?p > 39k N, Z(pk) = pk 1.p = 2, KkZ(2k) = 2k+1 1.(4)Z(n)\,=Z(m+n)uZ(m)+Z(n);Z(n), =Z(m
n)uZ(m) Z(n).l{5w, Z(n)5, 'u5ku
42
-
1n Smarandache9Smarandache'
?.
dSmarandache,Sdf(n)Sdf(n) = min{m : m N, n|m!!}. TSmarandacheV, 'u5, k?1
(J[4,6,31,100]. ~X, Sdf(n)k5,
=Sdf(n)5C, `en, KSdf(n), en
KSdf(n). Nuy, Sdf(n)5,c3n
Sdf(n)Ly5.
Z(n)Sdf(n), Ly5, U3m
XQ? !8e)5, =
Z(n) = Sdf(n), Z(n) + 1 = Sdf(n)
k).!L{0K), =yen.
nnn 3.6 ?n > 1,
Z(n) = Sdf(n) (3-12)
=k),k)=k2/:
n = 45;
n = p11 p22 pkk p,
k > 1, p1 < p2 < < pk < p, 1 6 i 6 k, i > 1,p + 1 0(modpii ).
nnn 3.7 ?n > 1,
Z(n) + 1 = Sdf(n) (3-13)
=k), k)=k3/:
n = 9;
n = p;
n = p11 p22 pkk p,
k > 1, p1 < p2 < < pk < p , 1 6 i 6 k, i > 0 3i > 1, p 1 0(modpii ).
eyn. kn3.6. , n = 1, Z(n) = Sdf(n).
Ny, n = 2, 3, 4, 5, 6, 7, Z(n) = Sdf(n). un > 8.
ky(3-12)Uk). n = 2p11 p12 pkk nIO).
(1)ek = 0,=n = 2, ddSdf(n)5Sdf(2), Z(2) = 2+1 1, lSdf(2) 6= Z(2).dnv.
43
-
Smarandache9'K
(2)ek > 1,Kn = 2p11 p12 pkk .dSdf(n) 5C, eZ(n) =
Sdf(n) = a,Ka7, Z(n) = Sdf(n) = 2m,udZ(n)k
2p11 p12 pkk |m(2m + 1). (3-14)
d(3-14), 7k2|m,pii 7kpii |m pi |(2m + 1). d2|m
2|m!!. (3-15)
PS1 = {pii : pii |m}, S2 = {pjj : p
jj |(2m + 1)}.
5,=S2 6= ,eS2 = ,Kkpii kpii |m,q2|m,ud2p11 p
22 pkk |
(2m1)2m2
, Z(n) 6 2m 1,S2 6= u:(i)eS1 = ,=pii , pii |(2m + 1). emax{pii } 6 m,((3-15), Sdf(n) 6 m;
emax{pii } > m,((3-15), Sdf(n) = 2s, 7pi0i0
S2,pi0i0 |s. ,m 6= s.l
Sdf(n) 6= 2m;
(ii)eS1 6= , pii S1, pjj S2. emax{pii } < max{p
jj },aq(i)
{, max{pjj } 6 m,max{pjj } > m,kSdf(n) 6= 2m. emax{pii } >
max{pjj }, Sdf(n) 6 m.n(i),(ii), dSd(n) 6= Z(n).y(3-12)Uk).en, n = p11 p
12 pkk p nIO). e3?:
(1)ek = 0,=n = p,dZ(p) = p 1, Sdf(n), Z(p) 6=Sdf(p).
(2)ek > 1 = 1,Kn = p11 p12 pkk p.
emax{pii } < p, KdSdf(n)5
Sdf(n) = Sdf(p11 p12 pkk p) = max{Sdf(p11 ), , Sdf(p
kk ), Sdf(p)} = p.
eZ(n) = Sdf(n) = p, KdZ(n)
p11 p12 pkk p
p(p + 1)
2.
u7Lpii |(p + 1), =p + 1 0(modpii ).,, yvp + 1 0(modpii ), kZ(n) = p.h < p,(i)eh < p 1,Kdp h(h+1)
2, p11 p
12 pkk p
h(h+1)2
, Z(n) 6= h;(ii)eh = p1,Kdpii |(p+1)9(p1, p+1) = 2, p11 p12 pkk p
(p1)p2 ,Z(n) 6=
p 1.l, ep + 1 0(modpii ), KkZ(n) = Sdf(n), =(3-12)|).emax{pii } > p, KdSdf(n) = p, Sdf(n) = max
16i6k{Sdf(pii )} = Sdf(p
jj ).
(i)eSdf(n) = p,Kdmax{pii } > p, max{pii } p(p+1)
2, Z(n) 6= p,lZ(n) 6=
44
-
1n Smarandache9Smarandache'
Sdf(n).
(ii)eSdf(n) = max16i6k
{Sdf(pii )} = Sdf(pjj ), dSdf(n)5,7kpj |Sdf(p
jj ),
upj |Sdf(n).k5, =?p9 > 2, p = 32, kSdf(p) < p,
ep 6 2 1, KSdf(p) = (2 1)p < p(p = 32); ep < 2 1, KNdp2|p!!, Sdf(p) < p. Sdf(n) = max
16i6k{Sdf(pii )} = Sdf(32) = 9, N
yn = 32 5 = 45), {.uSdf(n) = max
16i6k{Sdf(pii )} = Sdf(p
jj ) < p
jj , eZ(n) = Sdf(n) =
m,Km < pjj p
11 p
12 pkk p|
m(m+1)2 . qd, pj |m,lp
jj |m,m 1 > 2, Kn = p11 p12 pkk p.
eSdf(n) = max16i6k
{Sdf(pii )} = Sdf(p), dk > 1p 6= 32,uSdf(n) =
Sdf(p) = m < p.ddp|m9m < pp m(m+1)2 , dZ(n) 6= m,Z(n) 6=Sdf(n).
eSdf(n) = max16i6k
{Sdf(pii )} = Sdf(pjj ), d > 2, 3ej >
2pjj 6= 32,uSdf(n) = Sdf(p
jj ) = m < p
jj . , dp
jj
m(m+1)2, Z(n) 6=
m,Z(n) 6= Sdf(n).uBn3.6y.
eyn3.7. n3.6y{q, VL. dZ(p) = p 19Sdf(p) = p,n = pv(3-13). n = pk/, Z(pk) + 1 = pk 1 + 1 = pk,n = 32Sdf(pk) < pk,,Uv(3-13). n = 32 = 9T
(3-13)A). n = p11 p12 pkk p/, Ny, Z(n) + 1 =
Sdf(n)=vp 1 0(modpii ). {n3.6y{aq,p2K.
3.6 a2Smarandache
SmarandacheZ(n):
Z(n) = min{m : m N+, n
m(m + 1)
2}. (3-16)
z[105]2Smarandache, ?n,2Smarandache
Z3(n)
Z3(n) = min{m : m N+, n
m(m + 1)(m + 2)
6}. (3-17)
45
-
Smarandache9'K
!0'uZ3(n)5, On = 2lp, n = 2lpk, n = 2l
3k, l, k N+, p > 3, Z3(n)/. ), 9CKk[29,36,107].
nnn 3.10 ?k, p, eZ3(kp) = m, Km7lp 2, lp 1, lp, l N+n/.
yyy:Z3(n) > 1,p = 2, 3, nw,. p > 3, dp|m(m + 1)(m +2)/6, Kp7m, m + 1, m + 2, dm7klp 2, lp 1, lp, l N+n/.y..
e'uZ3(n)A{59y.
(1)n = 2lp, l N+, p > 3,Z3(2lp) = m,m = gp2,m = gp1,m =gpn9gk:
(a)eg = 1, m = p 2,d=km + 1 = p 1, d2lp|(p 2)(p 1)p/6,Kk2l+1|p 1,^{L=, p 1 0(mod2l+1),Z3(2lp) = p 2;em =p 1, d2lp|(p 1)p(p + 1)/6,Kk2l|p 12l|p + 1,=p 1 0(mod2l),p 1 60(mod(2l+1)), Z3(2
lp) = p 1;e2lp|p(p + 1)(p + 2)/6,Kkp + 1 0(mod(l + 1)), dp + 1 0(mod2l),AkZ3(2lp) = p 1.
(b)n,eg = 2, m = 2p2,d2lp|(2p2)(2p1)2p/6,K2l1|p1,=p1 0(mod2),p 1 6 0(mod2l), Z3(2lp) = 2p 2;m 6= 2p 1,em = 2p 1,K2lp|(2p1)2p(2p+1)/6,U;em = 2p,d2lp|2p(2p+1)(2p+2)/6,K2l1|p+1,=p + 1 0(mod2l1),p + 1 6 0(mod2l).
(c)eg = 2k, k > 1, k N ,d2lp|(2kp 2)(2kp 1)2kp/6, K2l|2k,dZ3(n)5,2k = 2l, d, Z3(2
lp) = 2lp 2.(d)eg > 3, m = gp 2,d2lp|(gp 2)(gp 1)gp/6, k2l+1|gp 1;em =
gp 1,d2lp|(gp 1)gp(gp + 1)/6,K2l1|gp 12l|gp + 1;em = gp,d2lp|gp(gp +1)(gp + 2)/6, K2l+1|gp + 1,k2l|gp + 1, dAkm = gp 1.n, eg > 3,m = 2gp 2,dk2l1|gp 1;em = 2gp, g > 3, dk2l1|gp + 1.dmv
46
-
1n Smarandache9Smarandache'
kU/. n, k
Z3(2lp) =
p 2, p 1 0(mod2l+1);p 1, p 1 0(mod2l);2p 2, p 1 0(mod2l1);2p, p + 1 0(mod2l1);gp 2, gp 1 0(mod2l+1);gp 1, gp 1 0(mod2l);2sp 2, sp 1 0(mod2l1);2sp, sp + 1 0(mod2l1);2lp 2, ,
3 6 g 6 2l 1, 3 6 s 6 2l1 1.(2)n, n = 2lpk, l N+, k N+, p > 3, k
Z3(2lpk) =
pk 2, pk 1 0(mod2l+1);pk 1, pk 1 0(mod2l);2pk 2, pk 1 0(mod2l1);2pk, pk + 1 0(mod2l1);gpk 2, gpk 1 0(mod2l+1);gpk 1, gpk 1 0(mod2l);2spk 2, spk 1 0(mod2l1);2spk, spk + 1 0(mod2l1);2lpk 2, ,
3 6 g 6 2l 1, 3 6 s 6 2l1 1.n = 2l 3k, l, k N+, 9z[105]52 (z[105][4])k
Z3(2l 3k) =
3k+1 2, 3k+1 1 0(mod2l+1);3k+1 1, 3k+1 1 0(mod2l);2 3k+1 2, 3k+1 1 0(mod2l1);2 3k+1, 3k+1 + 1 0(mod2l1);g 3k+1 2, g 3k+1 1 0(mod2l+1);g 3k+1 1, g 3k+1 1 0(mod2l);2s 3k+1 2, s 3k+1 1 0(mod2l1);2s 3k+1, s 3k+1 + 1 0(mod2l1);2l 3k+1 2,,
47
-
Smarandache9'K
3 6 g 6 2l 1, 3 6 s 6 2l1 1.
3.7 Smarandache91aSmarandache
?n, SmarandacheS(n)mn|m!.=S(n) = min{m : m N, n|m!}.1aSmarandacheZ2(n)knk
2(k+1)2
4 ,
Z2(n) = min
{
k : k N, n
k2(k + 1)2
4
}
,
NLk8. 9k'Smarandache
z[7]. lS(n)9Z2(n)NcA:
S(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, .
Z2(1) = 1, Z2(2) = 3, Z2(3) = 2, Z2(4) = 3, Z2(5) = 4, Z2(6) = 3, Z2(7) = 6, Z2(8) = 7, .
'uS(n)5, N?1, k(J[28,36,97100]. ~
Xz[28]
S(m1 + m2 + + mk) =k
i=1
S(mi)
)5, |^)nny?k > 3, Tk
|)(m1, m2, , mk).z[36]S(n)K, yC
n6x
(S(n) P (n))2 = 2(32)x
32
3 ln x+ O
(
x32
ln2 x
)
,
P (n)Lnf,(s)LRiemann zeta-.
z[97 98]S(2p1(2p 1))e.OK, y?p > 7, kO
S(2p1(2p 1)) > 6p + 19S(2p + 1) > 6p + 1.
C, z[99](:=y?p > 17?
a 9b,kO
S(ap + bp) > 8p + 1.
d, z[100]?Smarandache,e.OK, =Smarandache
48
-
1n Smarandache9Smarandache'
e.OK, y?n > 3kO:
S(Fn) = S(22n + 1) > 8 2n + 1,
Fn = 22n + 1.
'uS(n)SN~, p2. uZ2(n)5,
8), $kC5.
!8|^{Z2(n) + 1 = S(n))5,
k), N/`ye:
nnn 3.8 ?n,Z2(n) + 1 = S(n)k=ken/):
(A)n = 3, 4, 12, 33, 2 33, 22 33, 23 33, 24 33, 34, 2 34, 22 34, 23 34, 24 34;(B)n = p m,p > 5, m (p1)2
4?;
(C)n = p2 m,p > 52p 1, m(2p 1)2?u1.
w,Tn.)Z2(n) + 1 = S(n))5K. =y
k)z)N/.
e|^{9Smarandache5ny. k'g,
59k'SNz[4 6]. Ny:
n = 3, 4, 12, 33, 2 33, 22 33, 23 33, 24 33, 34, 2 34, 22 34, 23 34, 24 34
, Z2(n) + 1 = S(n)w,. y3eA[?:
S(n) = S(p) = p > 5, n = mp,KS(m) < p(m,p) = 1.denv
Z2(n)+1 = S(n),@oZ2(n) = p1,dZ2(n)kn (p1)2p2
4,=mp| (p1)2p2
4.
m| (p1)24 ,dm(p1)2
4 ?. , n = mpm|(p1)2
4 , kS(n) =
p, Z2(n) = p 1, n = mpZ2(n) + 1 = S(n)). uyn1(B).
S(n) = S(p2) = 2p, p > 5, n = mp2,KS(n) < 2p(m, p) = 1.denv
Z2(n) + 1 = S(n),@oZ2(n) = 2p 1,dZ2(n)kn (2p 1)2p2,
mp2|(2p 1)p2,m|(2p 1)2.
w,m 6= 1, Kn = p2, S(p2) = 2p,Z2(p2) = p 1.dn = p2vZ2(n) + 1 = S(n).um7L(2p 1)2u1. d, 2p 1, n = mp2, Z2(n) 6= p 1, Z2(n) 6= p, Z2(n) = 2p 1, S(n) = 2p, n = mp2 vZ2(n) + 1 = S(n).uyn/(C).
y3yS(n) = S(p)p > 59 > 3, nvZ2(n) + 1 = S(n).
n = mp, S(m) 6 S(p), (m, p) = 1. ukS(n) = hp, ph 6 .env
49
-
Smarandache9'K
Z2(n) + 1 = S(n),KZ2(n) = hp 1. udZ2(n)k
n = mp
(hp 1)2m2p24
ld5p2|h2 6 2.p|h. > h > 5. p > 5 > 5,p2|h2 6 2U, dkp2 > 2.y3S(n) = 3, dn = 36.yn = 3Z2(n) + 1 = S(n)
); S(n) = S(32) = 6, n = 9, 18, 36, 45, yvZ2(n) + 1 =
S(n);S(n) = S(33) = 9,
n = 33, 2 33, 22 33, 5 33, 7 33, 23 33, 10 33, 14 33, 16 33, 20 33.
dy
n = 33, 2 33, 22 33, 23 33, 24 33
vZ2(n) + 1 = S(n);S(n) = S(34) = 9, k
n = 34, 2 34, 22 34, 23 34, 24 34
vZ2(n) + 1 = S(n).
S(n) = S(2).w,n = 1, 2vZ2(n)+1 = S(n). eS(n) = S(4) =
4, @on = 4, 12.yn = 4, 12vZ2(n) + 1 = S(n);S(n) = S(23) = 4, d
n = 8, 24.ynvZ2(n) + 1 = S(n);S(n) = S(24) = 6, d
n = 16, 3 16, 5 16, 15 16.uvZ2(n) + 1 = S(n); S(n) =S(25) = 8,
n = 25, 3 25, 5 25, 7 25, 15 25, 21 25, 35 25, 105 25,
dNyvZ2(n) + 1 = S(n);
S(n) = S(2) = 2h, > max{6, h}
, n = m 2, KS(m) < S(2)(m, 2) = 1.denvZ2(n) + 1 = S(n), KdZ2(n),
n = m 2
(2h 1)2(2h)24
= (2h 1)2h2 ,
dd2|h2 6 ( 1)2 < 2, 95U, A^8B{Ny > 6,2 > ( 1)2 > h2.n, n3.8y.
50
-
1n Smarandache9Smarandache'
3.8 'uSmarandache LCMSmarandache
?n,SmarandacheS(n)mn|m!.=S(n) = min{m : m N, n|m!}, Smarandache LCMSL(n)k,n|[1, 2, , k],[1, 2, , k] L1, 2, , k. 'u5, N?1, (J[8,14,29,3739] . ~X, z[39]
SL(n)K, yC
n6x
(SL(n) P (n))2 = 25 (
5
2 x
52
ln x
)
+ O
(
x52
ln x
)
,
P (n)Lnf.
z[39]?SL(n) = S(n))5, )TK. =y:?
vTLn = 12n = p11 p22 prr p, p1, p2, , pr, p
1, 2, , r, vp > pii (i = 1, 2, , r).!0|^9|{
n6x
S(n)
SL(n)(3-18)
C5. Kk, (3-18)C5N
55, XJC
n6x
S(n)
SL(n) x
, @oS(n)SL(n)A??!!K?1
, y(5. N/`=ye(.
nnn 3.9 ?x > 1kC
n6x
S(n)
SL(n)= x + O
(x ln ln x
ln x
)
.
nnn 3.10 ?x > 1kC
n6x
P (n)
SL(n)= x + O
(x ln ln x
ln x
)
,
P (n)Lnf.
w,n~f, == ln ln xlnx f, 3
rCkK.
51
-
Smarandache9'K
eyn. yn3.9, aq/, n3.10. L{
C/
n6x
S(n)
SL(n)=
n6x
S(n) SL(n)SL(n)
+
n6x
1
=x + O
n6x
|SL(n) S(n)|SL(n)
. (3-19)
y3|^S(n)9SL(n)59|{5O(3-19).
dSL(n)5nIO)p11 p22 pkk k
SL(n) = max{p11 , p22 , , pkk }.
eSL(n)p,@oS(n)p.d, 3ekSL(n) S(n) = 0.3(3-19), k"[email protected](n)un, =
SL(n) = max{p11 , p22 , , pkk } p, > 2.
Am[1, x]kv^n8, ?n A, n = p11 , p22 , , pkk =p n1, (p, n1) = 1. y3?:A = B + C,n BXJSL(n) =p > ln
2 x9(ln lnx)2
. n CXJSL(n) = p < ln2 x9(ln lnx)2
. uk
n6x
|SL(n) S(n)|SL(n)
=
nB
|SL(n) S(n)|SL(n)
+
nC
|SL(n) S(n)|SL(n)
6
n69x(ln ln x)2
ln2 x
ln2 x9(ln ln x)2
6p6 xn
>2
1 +
nC1 R1 + R2. (3-20)
y3OO(3-20). kOR1. 5p 6 ln4 xk 6 4 ln ln x u
dnk
R1 6
n6 xln4 x
p6 xn>2
1 +
xln4 x
6n6 9x(ln ln x)2
ln2 x
p6 xn>2
n6 xln4 x
p6
xn
6ln x
1 +
xln4 x
6n6 9x(ln ln x)2
ln2 x
p6
xn
64 ln lnx
1
n6 xln4 x
x
n
ln x
ln ln x+
xln4 x
6n6 9x(ln ln x)2
ln2 x
x
n
x ln ln xln x
. (3-21)
52
-
1n Smarandache9Smarandache'
y3OR2,58CLp11 p
22 pkk ,
i 6 2 ln ln x, pi 6lnx
3 ln lnx, i = 1, 2, . u5
p6y
ln p = y + O
(y
ln y
)
, ln(
1 1ln p
)
1ln p
,
k
R2 =
nC1 6
p6 ln x3 ln ln x
0662 ln ln x
p
=
p6 ln x3 ln ln x
p2 ln ln x
1 1ln p
p6 ln x3 ln ln x
(
1 1ln p
)1exp
2 ln ln x
p6 ln x3 ln ln x
ln p
exp
3
4ln x +
p6 ln x3 ln ln x
1
ln p
xln x
, (3-22)
exp(y) = ey.
((3-20),(3-21)9(3-22), O
n6x
|SL(n) S(n)|SL(n)
x ln ln xln x
. (3-23)
|^(3-19)9(3-23)C
n6x
S(n)
SL(n)= x + O
(x ln ln x
ln x
)
.
un3.9y.
5SL(n) = p,S(n) = P (n) = p; SL(n), P (n) 6
S(n) 6 SL(n), udyn3.9{n3.10.
3.9 'uSmarandache LCM9
u?n, Smarandache LCM
SL(n) = min{k : k|[1, 2, , k], k N}.
53
-
Smarandache9'K
'u59kN?1L,
(J. ~XMurthy3
z[14]yn,SL(n) = S(n). ?
SL(n) = S(n), S(n) 6= n
)5. pS(n) = min{m : n|m!,m N}Smaranache. Le Mao-hua3z[19] )Murthy3z[14]JK, =n = 12 n =
p11 p22 prr p, (p > pii , i = 1, 2, , r)k).Lv Zhongtian 3z[20]?SL(n)C5, O
n6x
SL(n) =2
12
x2
ln x+
k
i=2
ci x2lni x
+ O
(x2
lnk+1 x
)
,
px > 1, k,cO~.
u?n,Smarandache LCM
SL(n) = max{k : [1, 2, , k]|k, k N}.
dSL(n)N,SL(1) = 1, SL(2) = 2, SL(3) = 1, SL(4) = 2, SL(5) =
1, SL(6) = 3, SL(7) = 1, SL(8) = 2, SL(9) = 1, SL(10) = 2,. w,, n
, SL(n) = 1, n, SL(n) > 2. 'u5, N?
1L, XJ, z[4, 19 21].~X3z[21], Tian Chengliang
d|nSL(d) = n
d|nSL(d) = (n)
)5, ck)n = 1, )n = 1, 3, 14. d
3z[21], SL(n)?9, k(J:
n=1
SL(n)
ns= (s)
=1
p
(p1)(ps 1)[1, 2, , p]
n6x
SL(n) = c x + o(ln2 x),
(s)Riemann zeta-.
54
-
1n Smarandache9Smarandache'
Z3z[59]
d|n SL(d) =
d|n S(d), )
(1)n;
(2)3n, Kn = 2p11 p22 pkk , p1 > 5, > 1, i > 0, k > 1, i = 1, 2, , k;(3)3|n,Kn = 2 31p22 pkk , p2 > 5, > 1, i > 0, k > 2, i = 1, 2, , k.
A =
n :
d|nSL(d) =
d|nS(d), n N
,
f(s) =
n=1nA
1
ns, (Re(s) > 1),
KZ'u)8
f(s) = (s)
(
1 112s
)
,
(s)Riemann zeta-.
!8|^{
d|nSL(d) + 1 = 2(n) (3-24)
)5, k), =en.
nnn 3.11 ?n, (3-24)k)=n = p, > 1, p > 3
.
yyy: dSL(n), n = 1, 2w,(3-24)). e6?1?
:
(1)en = p > 3,K(n) = (p) = 1,
d|nSL(d)+1 = SL(1) SL(p)+1 = 2,
, w,n = p(3-24)).
(2)en, @oXe:
1)n = 2, > 1,(n) = (2) = 1,
d|nSL(d) + 1 = SL(1) SL(2) SL(2) + 1 = 2 + 1,
K2 + 1 = 2,=2 = 1, = 0, > 1g. n = 2, > 1(3-24)).
2)n = 2p, > 1, > 1, p > 3, d(n) = (2p) = 2,
55
-
Smarandache9'K
dSL(n)5
d|2pSL(d) + 1 =
m=0
n=0SL(2mpn) + 1 > 2+ + 1 > 4,
n = 2p , > 1, > 1, p > 3, (3-24)).
3)n = 2p11 p22 pkk , > 1, i > 1, i = 1, 2, , k, k > 1, pi > 3 p
.(n) = (2p11 p22 pkk ) = k + 1,
d|2p11 p22 p
kk
SL(d) + 1=
n=0
1
m1=0
k
mk=0
SL(2npm11 pm22 pmkk ) + 1
>2+
k
i=1i
+ 1
>2+k1 + 1
>2k+1 + 1
> 2k+1.
n = 2p11 p22 pkk , (3-24)).
4)n = p, > 1, p > 3. k(n) = (p) = 1,d
d|pSL(d) + 1 =
m=0SL(pm) + 1 = 1 + 1 = 2, (3-24). n = p
(3-24)).
5)n = p11 p22 pkk , i > 1, 3 6 p1 < p2 < pk, i = 1, 2, , k, k > 2, du
(n) = (p11 p22 pkk ) = k,
d|p11 p22 p
kk
SL(d) + 1 =1
m1=0
2
m2=0
k
mk=0
SL(pm11 pm22 pmkk ) + 1 = 1 + 1 = 2,
d7k2 = 2k,=k = 1,k > 2g.n = p11 p22 pkk , (3-24)).
n?(1)(2), (3-24)k)=n = p, > 1, p > 3,
n3.11y.
56
-
1o SmarandacheS
1o SmarandacheS
4.1 Smarandache LCM 'SI
(x1, x2, ..., xt)[x1, x2, ..., xt]OLx1, x2, ..., xtf.
r u1. ?n, K
T (r, n) =[n, n + 1, ..., n + r 1]
[1, 2, ..., r]
PSLR(r) = {T (r, n)}n=1rSmarandache LCM'S. Maohua Le [22]
5, 'uSLR(2), SLR(3) SLR(4)4, w,
T (2, n) =1
2n(n + 1).
T (3, n) =
16n(n + 1)(n + 2), en ;
112
n(n + 1)(n + 2),en .
T (4, n) =
124n(n + 1)(n + 2)(n + 3), en 6 0(mod3);172n(n + 1)(n + 2)(n + 3), en 0(mod3).
!r = 54, =en.
nnn 4.1 u?n,k
T (5, n) =
57
-
Smarandache9'K
1120
n(n + 1)(n + 2)(n + 3)(n + 4), n 6 0(mod2)n 6 0(mod3)n 6 0(mod4)n + 1 6 0(mod3);
1240n(n + 1)(n + 2)(n + 3)(n + 4), n 0(mod2)n 6 0(mod3)n 6 0(mod4)
n + 1 6 0(mod3);1
360n(n + 1)(n + 2)(n + 3)(n + 4), n 0(mod2), n 6 0(mod3)n 6 0(mod2),n 6 0(mod3), n 6 0(mod4), n + 1 6 0(mod3);
1480
n(n + 1)(n + 2)(n + 3)(n + 4), n 6 0(mod3), n 0(mod4), n + 1 6 0(mod3);1
720n(n + 1)(n + 2)(n + 3)(n + 4), n 0(mod2), n 6 0(mod3), n 6 0(mod4),n + 1 (mod3)n 0(mod2), n 0(mod3),n 6 0(mod4);
11440n(n + 1)(n + 2)(n + 3)(n + 4), n 6 0(mod3), n 0(mod4), n + 1 0(mod3)
n 0(mod3), n 0(mod4).
e|^55yn4.1. ,[n, n + 1, n + 2, n +
3, n + 4] = [n(n + 1), (n + 2)(n + 3), n + 4], [n(n + 1), (n + 2)(n + 3), n + 4](n(n +
1), (n+2)(n+3), n+4) 6= n(n+1)(n+2)(n+3)(n+4). u?52?n9n. )T (r, n)
4, '[n, n + 1, n + 2, n + 3, n + 4], 3dnOa?1`. du
{'Xd'X, ^r8yeZda.
(1)en 0(mod2), n 6 0(mod3), n 6 0(mod4),
[n, n+1, n+2, n+3, n+4]
2 3 4 5 6
10 11 12 13 14
14 15 16 17 18
22 23 24 25 26
26 27 28 29 30
34 35 36 37 38
38 39 40 41 42
n[n, n+2, n+4] = 14[n(n+2)(n+4)],[n+1, n+3] = (n+1)(n+3),
en + 1 0(mod3),Kn + 4 0(mod3),
[n, n + 1, n + 2, n + 3, n + 4] =
112
[n(n + 1)(n + 2)(n + 3)(n + 4)],n + 1 0(mod3),14 [n(n + 1)(n + 2)(n + 3)(n + 4)], n + 1 6 0(mod3).
58
-
1o SmarandacheS
(2)en 0(mod3),n 6 0(mod2),
[n, n+1, n+2, n+3, n+4]
3 4 5 6 7
9 10 11 12 13
15 16 17 18 19
21 22 23 24 25
27 28 29 30 31
33 34 35 36 37
[n + 1, n + 3] = 12 [(n + 1)(n + 3)], qn 0(mod3),n + 3 0(mod3),[n, n + 3] = 13 [n(n + 3)],d
[n, n + 1, n + 2, n + 3, n + 4] =1
6[n(n + 1)(n + 2)(n + 3)(n + 4)].
(3)en 0(mod4),n 6 0(mod3),
[n, n+1, n+2, n+3, n+4]
4 5 6 7 8
8 9 10 11 12
16 17 18 19 20
20 21 22 23 24
28 29 30 31 32
32 33 34 35 36
40 41 42 43 44
n, qn 0(mod4),n + 4 0(mod4), [n, n + 2, n + 4] = 18 [n(n +2)(n + 4)], en + 1 0(mod3),Kn + 4 0(mod3),
[n, n + 1, n + 2, n + 3, n + 4] =
18n(n + 1)(n + 2)(n + 3)(n + 4), n + 1 6 0(mod3),124
n(n + 1)(n + 2)(n + 3)(n + 4),n + 1 0(mod3).
(4)en 0(mod2),n 0(mod3),n 6 0(mod4),
59
-
Smarandache9'K
[n, n+1, n+2, n+3, n+4]
6 7 8 9 10
18 19 20 21 22
30 31 32 33 34
42 43 44 45 46
54 55 56 57 58
n[n, n + 2, n + 4] = 14 [n(n + 2)(n + 4)],n n + 3 0(mod3),
[n, n + 1, n + 2, n + 3, n + 4] =1
12[n(n + 1)(n + 2)(n + 3)(n + 4)].
(5)en 0(mod3),n 0(mod4),
[n, n+1, n+2, n+3, n+4]
12 13 14 15 16
24 25 26 27 28
36 37 38 39 40
48 49 50 51 52
60 61 62 63 64
n[n, n + 2, n + 4] = 18[n(n + 2)(n + 4)],, n n + 4 0(mod4),n
n + 3 0(mod3), [n, n + 3] = 13 [n(n + 3)],
[n, n + 1, n + 2, n + 3, n + 4] =1
24[n(n + 1)(n + 2)(n + 3)(n + 4)].
(6)en 6 0(mod2),n 6 0(mod3),n 6 0(mod4)
60
-
1o SmarandacheS
[n, n+1, n+2, n+3, n+4]
5 6 7 8 9
7 8 9 10 11
11 12 13 14 15
13 14 15 16 17
17 18 19 20 21
19 20 21 22 23
23 24 25 26 27
25 26 27 28 29
[n+1, n+3] = 12 [(n+1)(n+3)],en+1 0(mod3),[n, n+1, n+2, n+3, n+4] = 1
6[n(n+1)(n+2)(n+3)(n+4)];en+1 0( mod 3), [n, n+1, n+2, n+3, n+4] =
12 [n(n + 1)(n + 2)(n + 3)(n + 4)].
r8a?1n, ^48a, Bn4.1.
4.2 Smarandache LCM 'SII
!Smarandache LCM'SO3u2,3,4,54,
!Smarandache LCM'S, Smarandache LCM'S
O'ur!'un, =eA(J.
nnn 4.2 ?g,n, rkXe4
T (r + 1, n) =n + r
r + 1
([1, 2, ..., r], r + 1)
([n, n + 1, ..., n + r 1], n + r)T (r, n),
4.1 XJr + 1 n + r Kk{
T (r + 1, n) =n + r
r + 1T (r, n).
nnn 4.3 ?g,n, rk,4
T (r, n + 1) =n + r
n
(n, [n + 1, ..., n + r])
([n, n + 1, ..., n + r 1], n + r)T (r, n).
4.2 XJn n + r r < nKk{
T (r, n + 1) =n + r
n T (r, n);
61
-
Smarandache9'K
XJn n + r r > nKk
T (r, n + 1) = (n + r) T (r, n).
nnn 4.4 ?g,n, rk4
T (r + 1, n + 1)=n + r
n n + r + 1
r + 1 ([1, 2, ..., r], r + 1)([n + 1, ..., n + r], n + r + 1)
(n, [n + 1, ..., n + r])([n, n + 1, ..., n + r 1], n + r) T (r, n).
yAn, IeAn.
nnn 4.1 ?ab, k(a, b)[a, b] = ab.
nnn 4.2 ?s, ts < t, k
(x1, x2, ..., xt) = ((x1, ..., xs), (xs+1, ..., xt))
[x1, x2, ..., xt] = [[x1, ..., xs], [xs+1, ..., xt]].
n4.14.2yz[6].
e5yn4,2. kT (r, n) , n4.14.2, k
[n, n + 1, ..., n + r]= [[n, n + 1, ..., n + r 1], n + r]
=[n, n + 1, ..., n + r 1](n + r)([n, n + 1, ..., n + r 1], n + r) ,
[1, 2, , r + 1] = [[1, 2, , r], r + 1] = (r + 1)[1, 2, , r]([1, 2, , r], r + 1) ,
u'uT (r + 1, n)4
T (r + 1, n)=[n, n + 1, ..., n + r]
[1, 2, ..., r + 1]
=[[n, n + 1, ..., n + r 1], n + r]
[[1, 2, ..., r], r + 1]
=
(n+r)[n,n+1,...,n+r1]([n,n+1,...,n+r1],n+r)
(r+1)[1,...,r]([1,2,...,r],r+1)
=n + r
r + 1
[n, n + 1, ..., n + r 1][1, 2, ..., r]
([1, 2, ..., r], r + 1)
([n, n + 1, ..., n + r 1], n + r)
=n + r
r + 1
([1, 2, ..., r], r + 1)
([n, n + 1, ..., n + r 1], n + r)T (r, n).
n4.2y.
62
-
1o SmarandacheS
4.1y. r + 1 n + r , w,k
([1, 2, , r], r + 1) = 1,
([n, n + 1, , n + r 1], n + r) = 1,
Kdn4.2, k
T (r + 1, n)=n + r
r + 1
([1, 2, ..., r], r + 1)
([n, n + 1, ..., n + r 1], n + r)T (r, n)
=n + r
r + 1T (r, n).
y4.1.
n4.3y: T (r, n)n4.1n4.25yn4.3. kd
n4.1n4.2, k
[n, [n + 1, n + 2, , n + r]] = n[n + 1, n + 2, , n + r](n, [n + 1, n + 2, , n + r]) ,
u
[n + 1, n + 2, , n + r]
=[n, [n + 1, n + 2, , n + r]] (n, [n + 1, n + 2, , n + r])
n
=[n, n + 1, n + 2, , n + r] (n, [n + 1, n + 2, , n + r])
n,
u'uT (r, n + 1)4
T (r, n + 1)=[n + 1, ..., n + r]
[1, 2, ..., r]
=[n, n + 1, ..., n + r](n, [n + 1, ..., n + r])
n
1
[1, 2, ..., r]
=(n, [n + 1, ..., n + r])
n[1, 2, ..., r]
[n, n + 1, ..., n + r 1](n + r)([n, n + 1, ..., n + r 1], n + r)
=n + r
n
(n, [n + 1, ..., n + r])
([n, n + 1, ..., n + r 1], n + r)[n, n + 1, ..., n + r 1]
[1, 2, ..., r]
=n + r
n
(n, [n + 1, ..., n + r])
([n, n + 1, ..., n + r 1], n + r)T (r, n).
n4.3y.
4.2y. nr < n, [n+1, n+2, , n+ r]7kfnKk
(n, [n + 1, n + 2, , n + r]) = 1;
63
-
Smarandache9'K
, en + r , K7,k
([n, n + 1, , n + r 1], n + r) = 1.
d
T (r, n + 1) =n + r
n
(n, [n + 1, ..., n + r])
([n, n + 1, ..., n + r 1], n + r)T (r, n) =n + r
nT (r, n).
nr > n, [n + 1, n + 2, , n + r] 7fn, d
(n, [n + 1, n + 2, , n + r]) = n,
dun + r , dk
T (r, n + 1)=n + r
n
(n, [n + 1, ..., n + r])
([n, n + 1, ..., n + r 1], n + r)T (r, n)
= (n + r)T (r, n).
y4.2.
n4.4y: A^n4.2n4.3 Nn4.4(J, =
T (r + 1, n + 1)
=n + r + 1
r + 1
([1, 2, ..., r], r + 1)
([n + 1, ..., n + r], n + r + 1)T (r, n + 1)
=(n + r + 1)(n + r)
(r + 1)n
([1, 2, ..., r], r + 1)
([n + 1, ..., n + r], n + r + 1)
(n, [n + 1, ..., n + r])
([n, ..., n + r 1], n + r)T (r, n).
n4.4y.
4.3 Smarandache 1
4.3.1 Smarandache1
u?n,n n1
1 2 n 1 n2 3 n 1...
......
...
n 1 n n 3 n 2n 1 n 2 n 1
(4-1)
nSmarrandache1, PSCND(n).
64
-
1o SmarandacheS
a,dE,n n1
a a + d a + (n 2)da + (n 1)da + d a + 2d a + (n 1)d a
......
......
a + (n 2)da + (n 1)d a + (n 4)da + (n 3)da + (n 1)d a a + (n 3)da + (n 2)d
(4-2)
'u(a, d)nSmarrandache?1LSCAD(n; a, d).
3z[34], Murthyen.
nnn 4.3 u?n,
SCND(n) = (1)n2 nn1 (n + 1)2
. (4-3)
nnn 4.4 u?n?Ea, d, k
SCAD(n; a, d) =
a, n = 1,
(1)n2 (nd)n1 (a+(n1)d)2 ,n > 1.(4-4)
3z[35], Le Maohuay'uSCND(n)SCAD(n; a, d). =Le
Maohua|^1
a1 a2 an1 anan a1 an2 an1...
......
...
a3 a4 a1 a2a2 a3 an a1
=
xn=1
(a1 + a2x + + anxn) (4-5)
yn4.34.4SO=.
XO(4-1), l.1n1m, 11\e1, 11,
SCND(n) =
1 2 n 1 n1 1 1 1 n...
......
...
1 1 1 111 n 1 1
.
65
-
Smarandache9'K
,121n\1, 2r1m(,=
SCND(n)=
n(n 1)/2 2 n 1 n0 1 1 1 n...
......
...
0 1 1 10 1 n 1 1
=n(n + 1)
2
1 11 n...
......
1 1 11 n 1 1
n1
=(1)n2 nn1 (n + 1)2
.
, (4-2)^{O.
rSCND(n)z, a1, a2, , annE,n n 1
a1 a2 an1 ana2 a3 an a1...
......
...
an1 an an3 an2an a1 an2 an1
(4-6)
'ua1, a2, , annSmarandache1LSCD(a1, a2, , an).unSmarandache1SCD(a1, a2, , an), A^(4-5), en.
nnn 4.5 un?Ea1, a2, , an,
SCD(a1, a2, , an) = (1)r
xn=1
(a1 + a2x + + anxn1), (4-7)
r =
n2 1, en,(n 1)/2,en.
66
-
1o SmarandacheS
SCD(a1, a2, , an),A: a, qE, n n1
a aq aqn2 aqn1
aq aq2 aqn1 a...
......
...
aqn2 aqn1 aqn4 aqn3
aqn1 a aqn3 aqn2
(4-8)
'u(a, q)nSmarandacheA?1LSCGD(n; a, q).
u, nSmarandacheA?1SCGD(n; a, q), kXen:
nnn 4.6 u?n?Ea, q,
SCGD(n; a, q) = (1)ran(1 qn)n1. (4-9)
n4.6y:l(4-6)!(4-8)k
SCGD(n; a, q) = SCD(a, aq, aq2, , aqn1),
2d(4-7),
SCD(a, aq, aq2, , aqn1)=(1)r
xn=1
(a + aqx + aq2x2 + + aqn1xn1)
=(1)ran
xn=1
(1 + qx + q2x2 + + qn1xn1).
XJxn = 1,K(1 + qx + q2x2 + + qn1xn1)(1 qx) = 1 qn,du
xn=1
(1 qx) = qn
xn=1
(1
q x)
= qn(
1
qn x)
= 1 qn,
(4-9). n4.6. S, n4.3!n4.4!n4.6n4.5A
.
4.3.2 Smarandache V1
3z[35], u?n, n n1
1 2 n 1 n2 3 n n 1...
......
...
n 1 n 3 2n n 1 2 1
(4-10)
nSmarandacheV1LSBND(n).
67
-
Smarandache9'K
a,bE,n n1
a a + d a + (n 2)da + (n 1)da + b a + 2d a + (n 1)da + (n 2)d
......
......
a + (n 2)da + (n 1)d a + 2d a + da + (n 1)da + (n 2)d a + d a
(4-11)
'u(a, d)nSmarandacheV?1LSBAD(n; a, d).
'unSmarandachV1SBND(n)SmarandachV?1
SBAD(n; a, d) , Le Maohuayen4.6,n4.7.
nnn 4.5 u?n,
SBND(n) = (1)n(n1)
2 2n1(n + 1). (4-12)
nnn 4.6 u?n?Ea, d,
SBAD(n; a, d) = (1)n(n1)
2 2n2dn1 (2a + (n 1)d) . (4-13)
a, dE, n n1
a aq aqn2 aqn1
aq aq2 aqn1 aqn2...
......
...
aqn2 aqn1 aq2 aqaqn1 aqn2 aq a
(4-14)
'u(a, q)nSmarandacheVA?1LSBGD(n; a, q).
'unSmarandacheVA?1SBGD(n; a, q),ken.
nnn 4.7 u?nEa, d,
SCGD(n; a, d) =
0, n = 2,
(1)n(n1)2 anqn(n2)(n1)(q2 1)n1,n 6= 2.(4-15)
n4.7y:O(4-14), IO
1 q an2 qn1
q q2 qn1 qn2...
......
...
qn2 qn1 q2 qqn1 qn2 q 1
. (4-16)
68
-
1o SmarandacheS
d(4-14), n = 1n = 2, (4-15). dbn > 2. r(4-16)
12Jfq,12 1\11, ,U11m,
1 q an2 qn1
q q2 qn1 qn2...
......
...
qn2 qn1 q2 qqn1 qn2 q 1
= q(1)n+1(qn1 qn3)
1 q an2 qn1
q q3 qn1 qn2...
......
...
qn3 qn1 q3 q2
qn2 qn2 q2 q
= q(1)n1(qn1 qn3)q2(1)n (qn2 qn4)
qn2(1)4(q2 1)
1 qn1
q qn2
=(1)n(n1)
2 q(n2)(n1)(q2 1)n1.
dd, (4-15). n4.7y.
2SBND(n)/. a1, a2, , an,n E, n n1
a1 a2 an1 ana2 a3 an an1...
......
...
an1 an a3 a2an an1 a2 a1
(4-17)
'ua1, a2, , annSmarandacheV1LSBD(a1, a2, , an).uSBD(a1, a2, , an)O, )K.
4.4 Smarandache
N+N8. un,
S(n) = min{k|k N+, n|k!}. (4-18)
XdS(n)'unSmarandache. n, XJn
u2n, Kn. 5,
-
Smarandache9'K
12. !?0Smarandache, =y:
nnn 4.8 =kSmarandache12.
nnn 4.7 XJ
n = pr11 pr22 prkk (4-20)
nIO), K
S(n) = max {S(pr11 ), S(pr22 ), , S(prkk )} . (4-21)
yyy:z[43].
nnn 4.8 upr, 7kS(pr) 6 pr.
yyy:z[43].
nnn 4.9 un,d(n)n. d,d(n)5;(4-20)nI
O),K
d(n) = (r1 + 1)(r2 + 1) (rk + 1). (4-22)
yyy:z[44]~6.4.2.
nnn 4.10
n
d(n)< 2, n N (4-23)
=k)n = 1, 2, 3, 4, 6.
eyn4.8. nn 6= 12Smarandache. z[41](J, n > 106. (4-20)nIO). n4.7
S(n) = S(pr), (4-24)
p = pj, r = rj, 1 6 j 6 k. (4-25)
l(4-24)
n|S(pr)!, (4-26)
un?dk
S(d) 6 S(pr). (4-27)
70
-
1o SmarandacheS
un,
g(n) =
d|nS(d). (4-28)
d, l(4-19)(4-28)n
g(n) = n + 1 + S(n). (4-29)
qd(n)n.l(4-27)(4-29)
d(n)S(pr) > n. (4-30)
dul(4-20)(4-24)gcd(pr, n/pr) = 1,
n = prm,m N, gcd(pr, m) = 1. (4-31)
qln4.9,d(n)5, l(4-31)
d(n) = d(pr)d(m) = (r + 1)d(m). (4-32)
(4-31)(4-32)\(4-30)=
(r + 1)S(pr)
pr> 3
m
d(m). (4-33)
f(n) = nd(n) , d(4-33)
(r + 1)S(pr)
pr> f(m). (4-34)
u, n4.8n4.10, l(4-34): =kSmarandache12. n4.8y..
4.5 Smarandache 3n iS
?n, Smarandache 3niSan
an = 13, 26, 39, 412, 515, 618, 721, 824, ....
Tz, 12113. ~X, a12 =
1236, a20 = 2060, a41 = 41123, a333 = 333999, ....;[Smarandache
3z[7]z[46]J,
-
Smarandache9'K
(b)n, anU;
(c)XJan,@okn = 22a1 32a2 52a3 112a4 n1,(n1, 330) = 1.
((5J, TKg{,
l,wSmarandache 3niSS35. 'ucn
Kk, `3a1 + a2 + ... + aN (
C? L{O, E,O, (JN(/
k', /n, lU, =UC. u,
!
ln a1 + ln a2 + ... ln aN
C5K, |^{9?5ye(.
nnn 4.9 ?N , kC
n6N
ln an = 2N ln N + O(N).
eyTn. kan(, nk, =n = bkbk1...b2b1,1 6
bk 6 9, 0 6 bi 6 9(i = 1, 2, ..., k 1). ud{?{K,
333...34
k16 n 6 333...3
k
, 3nk;
333...34
k
6 n 6 333...33
k+1
, 3nk + 1. dan
an = n (10k + 3),
an = n (10k+1 + 3).
?N , w,3M , :
333...33
M
< n 6 333...33
M+1
. (4-35)
udc, k
16n6N
an =3
n=1
an 33
n=4
an 13(10M1)
n= 13(10M11)+1
an N
n= 13(10M1)+1
an =
N !(10 + 3)3 (100 + 3)30 ... (10M + 3)310M1 (10M+1 + 3)N 13 (10M1). (4-36)
72
-
1o SmarandacheS
5x 0, kOln(1 + x) = x + O(x2),M
k=1
ln(10k + 3)310k1
=M
k=1
3 10k1 (
k ln 10 + 310k
+ O
(1
102k
))
=M
k=1
k 10k1 3 ln 10 + 910
M + O(1)
=1
3M 10M ln 10 1
27(10M 1) ln 10 + 9
10M + O(1)
=1
3M 10M ln 10 + O(N). (4-37)
ln(10M+1 + 3)N13(10
M1) =
(
N 13
(10M 1
))
ln(10M+1 + 3)
=
(
N 13
(10M 1
))
(M + 1) ln 10 + O(1)
=N M ln 10 13 M 10M ln 10 + O(N). (4-38)
A^Euler5, N
ln(N !) =
16n6N
ln n = N ln N N + O(1). (4-39)
5(4-35), JO
10M < N 6 10M+1,
ln N = M ln 10 + O(1). (4-40)
((4-36)9?(4-37)-(4-40), C
n6N
ln an
=
16n6N
ln n +M
k=1
ln(10k + 3)310k1
+ ln(10M+1 + 3)N13(10M1)
=2N ln N + O(N).
un4.9y. w,, C'o, 3(C, k
u?.
73
-
Smarandache9'K
4.6 Smarandache kn iS
uk N+, Smarandache kn iS{a (k, n)}8, T8z, 11k. ~X!
Smarandache 3niS
{a (3, n)} = {13, 26, 39, 412, 515, 618, 721, 824, ...} ,
=Tz, 113.
!'u
ln a (3, n)
5, yC
n6N
ln a (3, n) = 2N ln N + O (N) .
'u5, 8qvk
-
1o SmarandacheS
w,AC(, 3(CE,m
K, UY8I.
eyn. yn4.10 k = 293 , aq/
n4.10 k = 4, 59n4.11. kyn4.10k = 2. a (2, n)(, n
?Lk , =
n = bkbk1 b2b1,
1 6 bk 6 9, 0 6 bi 6 9, i = 1, 2, , k 1.
ud{?{K, 10k1 6 n 6 5 10k1 1 , 2n k ; 5 10k1 6n 6 10k 1, 2n k + 1. da (2, n)
a (2, n) = n (10k + 2
),
a (2, n) = n (10k+1 + 2
).
?x, w,M N+
5 10M 6 x < 5 10M+1, (4-41)
uk
16n6x
n
a (2, n)
=4
n=1
n
a (2, n)+
49
n=5
n
a (2, n)+
499
n=50
n
a (2, n)+
4999
n=500
n
a (2, n)+
+510M1
n=510M1
n
a (2, n)+
510M6n6x
n
a (2, n)
=4
n=1
1
10 + 2+
49
n=5
1
102 + 2+
499
n=50
1
103 + 2+
4999
n=500
1
104 + 2
+510M1
n=510M1
1
10M+1 + 2+
510M6n6x
1
10M+2 + 2
=5 110 + 2
+50 5102 + 2
+500 50103 + 2
+5000 500
104 + 2+
+ 5 10M 5 10M110M+1 + 2
+ O
(x 5 10M10M+2 + 2
)
=9 10
2 (102+2
) +9 102
2 (103+2
) +9 103
2 (104+2
) + + 9 10M
2 (10M+1+2
) + O (1)
75
-
Smarandache9'K
=9 (102 + 2 2
)
20 (102 + 2
) +9 (103 + 2 2
)
20 (103 + 2
) +9(104 + 2 2
)
20(104 + 2
) +
+ 9(10M+1 + 2 2
)
20(10M+1 + 2
) + O (1)
=9
20M + O
(M
m=1
1
10m
)
+ O (1)
=9
20M + O (1) . (4-42)
5(4-41) , kO
M ln 10 + ln 5 6 x < ln 5 + (M + 1) ln 10
M =1
ln 10ln x + O (1) .
d( 4-42) C
16n6x
n
a (k, n)=
9
20 ln 10 ln x + O (x) .
uyn4.10k = 2.
y3yn4.10 k = 3. a (3, n)(, n?L
k, =
n = bkbk1 b2b1,
1 6 bk 6 9, 0 6 bi 6 9, i = 1, 2, k 1.
ud{?{K,
333 34
k16 n 6 333 33
k
, 3nk;
333 34
k
6 n 6 333 33
k+1
, 3nk + 1. unk, da (3, n)
a (3, n) = n (10k + 3
),
a (3, n) = n (10k+1 + 3
).
76
-
1o SmarandacheS
?x, w,M N+,
333 33
M
6 x < 333 33
M+1
, (4-43)
uk
16n6N
n
a(3, n)
=3
n=1
n
a(3, n)+
33
n=4
n
a(3, n)+
333
n=34)
n
a(3, n)+
3333
n=334
n
a(3, n)+
13(10M1)
n= 13(10M11)+1
n
a(3, n)+
13(10M1)+16N6x
n
a(3, n)
=3
n=1
1
10 + 3+
33
n=4
1
102 + 3+
333
n=34
1
103 + 3+
3333
n=334
1
104 + 3+
+13(10M1)
n= 13(10M11)+1
1
10M + 3+
13(10M1)+16n6x
1
10M+1 + 3
=3
10 + 3+
33 3102 + 3
+333 33103 + 3
+3333 333
104 + 3+
+13 (10M 1) 1
3(10M1 1)
10M + 3+ O
(
x 13 (10M 1)
10M+1 + 3
)
=3 1
10 + 3+
3 10102 + 3
+3 102103 + 3
+3 103104 + 3
+
+ 3 10M1
10M + 3+ O(1)
=3 (10 + 3 3)10 (10 + 3) +
3 (102 + 3 3)10 (102 + 3) +
3 (103 + 3 3)10 (103 + 3) +
3 (104 + 3 3)10 (104 + 3) +
+ 3 (10M+1 + 3 3)
10 (10M + 3) + O(1)
=3
10M + O
(M
m=1
1
10m
)
+ O(1)
=3
10M + O(1). (4-44)
5(4-43) =
1
3(10M 1
)6 x 0, F0 = 0, F1 = 1, L0 = 2 L1 = 1. S3!
!OD)+uX4^. d'uFn Ln
5IS81. R.L.Duncan[76] L.Kuipers[78] O
31967c1969cylog Fn 1 . Neville Robbins[81] 31988c
/Xpx2 1, px3 1 (p )Fibonacci. 'uFibonacciSN, k(J. ~X,
m
n=0
Fn = Fm+2 1,
m1
n=0
F2n+1 = F2m,
m
n=0
nFn = mFm+2 Fm+3 + 2,
m
n=0
CnmFcnFnk F
mnk+1 = Fmk+c,
Cnm =m!
n!(mn)! .
2004c, +[84]Fn Ln A2
a1+a2++ak+1=nFm(a1+1)Fm(a2+1) . . . Fm(ak+1+1) = (1)mn
F k+1m2kk!
U(k)n+k
(im
2Lm
)
,
a1+a2++ak+1=n+k+1Lma1Lma2 . . . Lmak+1
82
-
1 K
= (1)m(n+k+1) 2k!
k+1
h=0
(im+2
2Lm
)h(k + 1)!
h!(k + 1 h)!U(k)n+2k+1h
(im
2Lm
)
,
k, m ?, n, a1, a2, . . . , ak+1 Ki 1 , Uk(x) 1aChebyshev.
w[90]3d:q'uFibonacciA
a1+a2++ak=nFa1+1(x) Fa2+1(x) Fak+1(x) =
[n2]
m=0
Cmn+k1m Ck1n+k12m xn2m,
FibonacciFn(x)d548Fn+2(x) = xFn+1(x) + Fn(x),
F0(x) = 0,F1(x) = 1; Cnm =
m!n!(mn)! ,[z]LLz; x = 1,
Fn(x) = Fn, ddA'uFibonacci,eA?k
n
a1+a2++ak=n+kFa1 Fa2 Fak =
[n2]
m=0
Cmn+k1m Ck1n+k12m,
a1+a2++ak=n+kF2a1 F2a2 F2ak = 3k 5
nk2
[n2]
m=0
Cmn+k1m Ck1n+k12m 5m,
a1+a2++ak=n+kF3a1 F3a2 F3ak = 22n+k
[n2]
m=0
Cmn+k1m Ck1n+k12m 16m,
a1+a2++ak=n+kF4a1 F4a2 F4ak = 3n 7k 5
nk2
[n2]
m=0
Cmn+k1m Ck1n+k12m 45m,
a1+a2++ak=n+kF5a1 F5a2 F5ak = 5k 11n
[n2]
m=0
Cmn+k1m Ck1n+k12m 121m.
!LrxnLChebyshev5FibonacciLucaso
. , yen.
nnn 5.1 ?Kn m, k
L2nm = (1)mn(2n)!
(n!)2+ (2n)!
n
k=1
(1)m(nk)(n k)!(n + k)!L2km;
L2n+1m = (2n + 1)!n
k=0
(1)m(nk)(n k)!(n + k + 1)!L(2k+1)m.
83
-
Smarandache9'K
nnn 5.2 ?Kn m, k
L2nm =(2n)!
Fm
n
k=0
(1)m(nk)(2k + 1)(n k)!(n + k + 1)!F(2k+1)m;
L2n+1m =2(2n + 1)!
Fm
n
k=0
(1)m(nk)(k + 1)(n k)!(n + k + 2)!F2(k+1)m.
!3nyI^An. 3nckI1
aChebyshevTn(x) 1aChebyshevUn(x) (n = 0, 1, ) L
Tn(x) =1
2
[(
x +
x2 1)n
+(
x
x2 1)n]
,
Un(x) =1
2
x2 1
[(
x +
x2 1)n+1
(
x
x2 1)n+1
]
.
,^Xe485:
Tn+2(x) = 2xTn+1(x) Tn(x),
Un+2(x) = 2xUn+1(x) Un(x),
n > 0, T0(x) = 1, T1(x) = x, U0(x) = 1 U1(x) = 2x.
eXeAn.
nnn 5.1 ?m n, k
Tn(Tm(x)) = Tmn(x),
Un(Tm(x)) =Um(n+1)1(x)
Um1(x).
y: z[84].
nnn 5.2 i 1 , m n ?, Kk
Un
(i
2
)
= inFn+1,
Tn
(i
2
)
=in
2Ln,
Tn
(
Tm
(i
2
))
=imn
2Lmn,
Un
(
Tm
(i
2
))
= imnFm(n+1)
Fm.
y: w,kUn(
i2
)= inFn+1, Tn
(i2
)= i
n
2 Ln, Kn5.1, qN/{e
. yn5.2.
84
-
1 K
nnn 5.3 ?Kn,
xn 12an0T0(x) +
k=1
ankTk(x) (5-1)
xn
k=0
bnkUk(x), (5-2)
Kk
ank =
2n!(nk)!!(n+k)!! , n > k,n + k;
0, .(5-3)
bnk =
2(k+1)n!(nk)!!(n+k+2)!! , n > k,n + k;
0, .(5-4)
y: , Chebyshevk5(z[92]). ~X
1
1
Tm(x)Tn(x)1 x2
dx =
0, m 6= n,2 , m = n > 0,
, m = n = 0;
(5-5)
Tn(cos ) = cosn; (5-6)
1
1
1 x2Um(x)Un(x) dx =
0, m 6= n,2 , m = n > 0,
, m = n = 0;
(5-7)
Un(cos ) =sin(n + 1)
sin . (5-8)
ky(5-3). ?Km, k(5-1)> Tm(x)1x2 , 2>
l1 1 , A^(5-5) 1
1
xnTm(x)1 x2
dx=1
2an0
1
1
Tm(x)T0(x)1 x2
dx +
k=1
ank
1
1
Tm(x)Tk(x)1 x2
dx
=
2anm, (m = 0, 1, 2, )
d,
anm =2
1
1
xnTm(x)1 x2
dx.
85
-
Smarandache9'K
x = cos t, (5-6)k
anm =2
0
cosnt cosmtdt
=2
0
cosnt (cos(m 1)t cos t sin(m 1)t sin t) dt
=2
0
cosn+1t cos(m 1)t dt 2
0
cosnt sin(m 1)t sin t dt
=an+1m1 +2
1n + 1
0
sin(m 1)t d(cosn+1t)
=an+1m1 +2
1n + 1
cosn+1t sin(m 1)t
0
2 m 1
n + 1
0
cosn+1t cos(m 1)t dt
=an+1m1 m 1n + 1
an+1m1
=n m + 2
n + 1 an+1m1
=n m + 2
n + 1 n m + 4
n + 2 an+2m2
=
=n m + 2
n + 1 n m + 4
n + 2 n m + 2m
n + m an+m0
=
(n+m)!!(nm)!!(n+m)!
n!
an+m0.
eOan+m0, w,
an+m0 =2
0
cosn+m t dt.
{z, In = 0
cosn t dt. n , f(x) = cosx 3m[0, ]
, K
In =
0
cosn t dt = 0; (5-9)
n , k
In =
0
cosn t dt =
0
cosn1 t d sin t
=sin t cosn1 t
0 + (n 1)
0
sin2 t cosn2 t dt
=(n 1)
0
(1 cos2 t
)cosn2 t dt
=(n 1)
0
cosn2 t dt (n 1)
0
cosn t dt
86
-
1 K
=(n 1)In2 + (n 1)In. (5-10)
d'uIn4
In =n 1
n In2. (5-11)
Kd(5-11)
In =n 1
n In2 =
n 1n
n 3n 2 In4
=
=n 1
n n 3n 2
1
2 I0, (5-12)
dIn
I0 =
0
cos0 t dt =
0
1 dt = . (5-13)
d((5-9), (5-12)(5-13),
In =
0
cosn t dt
=
(n1)!!n!!
, n ,
0, n .
dd
an+m0 =2
0
cosn+m t dt
=
2(n+m1)!!(n+m)!! , n + m ,
0, n + m .
d