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Five Properties of the Smarandache Double Factorial FunctionFelice Russo

Via A. Infante 767051 Avezzano Ag) Italyfelice.russo@Jcatamail.com

AbstractIn this paper some properties o the Smarandache double factorialfunction have been analyzed.

In [1], [2], [3J and [4] the Smarandache double fuctoriaI Sd.f(n) function is defined as the smallestnumber such that Sdf{n) is divisible by n, where the double factorial by definition is given by[6]:m = lx3x5x. .m, ifrn is oddm = 2x4x6x. ..m, ifrn is even.

In [2] several properties of that function have been analyzed n this paper five newproperties are reported

1. Sdf Pk+2) =p2 where p =2 k +1 is any prime and k any integerLet s consider the prime p =2k +1 . Then:

2 k 2 wh . .1 357 ......... ........3p .. ....... 5 ........... p =m p ere m IS any mteger.This because the number of terms multiples ofp up to p2 are k+ 1 and the last termcontains two times p.Then p2 is the least value such that 1 3 . 5 . 7 9 ..... p2 is divisible by pk 2.

2. Sdf P2) =3 p where p is any odd prime.

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In fact for any odd p we have:135 7 ......... p ...... 3p = m p2 where m is any integer.

i IOn -1 J j on -13. S4lk 9 =S4l 9 where n is any integer >1 and k=3,5,7,9i On -1Let s suppose that S4 l = m then:Ion -1 h . . B th .357 .......m=o w ere a IS any mteger. ut m e preVIOUS

mUltiplication there are factors multiple of3,5,7 and 9 and then:

I oJ. 5 7 ...m = dt - 10: I where a is any integer and iF3,5,7,9. Then:

i IOn -1 J /If IOn -1 J. S4 l 9 =S4 l2 . 9 where n is any integer>1 and k=2,4,6,8

( On -1 h . . B th .2468 ......m =0 2 9 w ere a IS any mteger. ut m e preVIous84

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multiplication there are factors multiple of4 6 and 8 and then:

(On -124 68 .....m a 2 k 9 where at is any integer and k=4,6,8.Then:

5. Sdj pm)= 2.m-1).p for p ~ 2 m - 1 ) . Here m is any integer and p anyodd prime.

This is a generalization ofproperty number 2 reported above.

References[1] C. Dumitrescu and V. Seleacu, Some notions nd questions in Number Theory, Erhus Univ.

Press, Glendale 1994:h t t o : l w w w . g : l l h m . l l n m . e d u / - s m 1 ~ r : m d a c h e / S N A O i : - . r T . t x t

[2] Felice Russo, A set of new Smarandache functions, sequences and conjectures in numbertheory, American Research Press, 2000:http://W\V\V.Q:2cllup. unm.edu/-sITIarandache/F dice-R.usso-book I. pdf[3] F. Smarandache, Only Problems, not Solutions , Xiquan, Publishing House, PhoenixClricago 1990, 1991, 1993.[4] F. Smarandache, he Florentin Smarandache papers special collection, Arizona StateUniversity, Hayden Library, Tempe, Box 871 006, AZ 85287-1006, USA[5] F. Smarandache, Collected Papers, VoL n KishinevUniversityPress, 1997.[6] E. Weisstein, eRe Concise Encyclopedia ofMathematics, eRe Press, 1999.

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