a binary sequence generator based on the kolakoski sequence and multiples of odd primes
DESCRIPTION
Kamawotan, vol. 1, no. 2, May 2007, pp. 79-87.TRANSCRIPT
KA
MA
WO
TA
N v
ol.
1,
no. 2
, M
ay
20
07
78
Sec
on
dary
So
urce
s
Ber
nasc
oni,
R.
(198
7).
De
cons
truc
tion
and
the
pos
sibi
lity
of e
thic
s. I
n J.
S
alli
s (E
d.),
Dec
ons
truc
tion
an
d p
hilo
sop
hy
. C
hica
go:
Uni
vers
ity o
f C
hica
go P
ress
.
Ca
puto
, J.
(19
93a
). Ag
ains
t e
thic
s: C
on
trib
utio
ns t
o a
po
etic
s of
ob
liga
tion
with
con
sta
nt r
efe
ren
ce t
o d
eco
nstr
uct
ion
, S
tudi
es i
n C
ontin
enta
l T
hou
ght.
Blo
omin
gton
: In
dia
na U
nive
rsity
Pre
ss.
Ca
puto
, J.
(19
93b)
. Ra
dica
l he
rme
neu
tics:
Re
petit
ion
, de
con
stru
ctio
n a
nd
the
her
me
neu
tic p
roje
ct, S
tudi
es in
Phe
nom
enol
ogy
and
Exi
sten
tial
Phi
loso
phy.
Blo
omin
gton
: In
dia
na U
nive
rsity
Pre
ss.
Ca
puto
, J.
(1
997a
). De
con
stru
ctio
n
in
a
nut
shel
l.
New
Y
ork:
F
ordh
am
Uni
vers
ity P
ress
.
Ca
puto
, J.
(19
97b)
. Th
e p
raye
rs a
nd
te
ars
of J
acq
ues
De
rrid
a:
Rel
igio
n w
itho
ut r
elig
ion.
Ind
iana
polis
: Ind
iana
Uni
vers
ity P
ress
.
Cor
nell,
D.
(Ed.
). (
1992
). De
cons
tru
ctio
n a
nd t
he
pos
sib
ility
of
just
ice
. N
ew
Yor
k: R
outle
dge.
Cri
tchl
ey,
S.
(199
0).
Der
rida
a
nd
deco
nstr
uctio
n.
In
R.
Kea
rne
y (E
d.),
R
ou
tledg
e h
isto
ry o
f p
hilo
sop
hy (vo
l. V
II).
Lond
on a
nd N
ew Y
ork:
R
outle
dge.
Cri
tchl
ey,
S.
(199
2). T
he e
thic
s of
de
con
stru
ctio
n: D
errid
a a
nd
Lev
ina
s.
Oxf
ord:
Bla
ckw
ell.
Cul
ler,
J. (
1983
). On
dec
ons
truc
tion. L
ondo
n: R
outle
dge
and
Keg
an
Pa
ul.
Kea
rne
y, R
. (19
99).
Po
etic
s of
mo
der
nity. N
ew Y
ork:
Hum
anity
Boo
ks.
Levi
nas,
E.
(196
9). T
ota
lity
and
inf
inity
. (A
. Li
ngis
, T
rans
.).
Pitt
sbur
gh:
Duq
uesn
e U
nive
rsity
Pre
ss.
Nor
ris, C
. (1
987)
. De
rrid
a. C
am
brid
ge:
Ha
rva
rd U
nive
rsity
Pre
ss.
Ra
papo
rt,
H. (
2003
). La
ter
Der
rida.
New
Yor
k: R
outle
dge.
Sim
, S. (
1992
). Bey
on
d a
esth
etic
s. Lon
don:
Ha
rves
ter
Whe
ats
hea
f.
Col
lege
of
Art
s a
nd
Sci
en
ces,
Ate
neo
de
Na
ga
Un
ive
rsi
ty
79
A B
inar
y S
eque
nce
Gen
erat
or B
ased
on
the
Kol
akos
ki S
eque
nce
and
Mu
ltip
les
of O
dd
Pri
mes
Joel
R. N
oche
D
epa
rtm
ent
of M
ath
em
atic
s a
nd N
atur
al S
cien
ces
A
bstr
act
An
ear
lier
bin
ary
seq
uen
ce g
ene
rato
r is
mo
difi
ed
to
yie
ld
be
tter
per
form
an
ce i
n t
he D
IEH
AR
D t
ests
for
ran
dom
ne
ss
of
un
iform
ly
dis
trib
ute
d
num
ber
s.
T
he
n
ew
g
ene
rato
r in
vert
s e
very
p
bits
of
a
mo
difi
ed
K
ola
kosk
i se
qu
ence
, w
he
re p
is a
prim
e n
um
ber
, a
nd d
oes
this
for
the
fi
rst f
od
d p
rim
e
num
ber
s.
S
amp
les
from
th
e
resu
ltin
g
bina
ry
sequ
enc
e c
an
be
use
d i
n s
cie
ntifi
c a
ppl
ica
tions
re
qui
rin
g ra
nd
om n
umbe
rs.
Intr
oduc
tion
Larg
e a
mo
unt
s of
“ra
ndo
m”
num
ber
s ar
e cu
rren
tly n
eed
ed
by
ma
ny
ap
plic
atio
ns (
Hay
es,
20
01)
. I
n th
is w
ork,
num
ber
s a
re s
aid
to
be
rand
om
if t
hey
perf
orm
wel
l in
com
mo
nly
acc
ept
ed
sta
tistic
al t
ests
of
ra
nd
om
nes
s.
Pse
ud
o-ra
ndo
m n
um
ber
ge
nera
tors
(P
RN
Gs)
usu
ally
re
fer
to g
ene
rato
rs o
f u
nifo
rmly
dis
trib
uted
nu
mb
ers
(oft
en o
ver
the
in
terv
al
(0,1
)).
(A
rbitr
ary
dist
rib
utio
ns
are
us
ually ge
nera
ted
by
app
lyin
g tr
ans
form
atio
ns t
o th
ese
valu
es.)
P
RN
Gs
diffe
r fr
om
(tr
ue)
rand
om
n
um
ber
g
ener
ato
rs
(RN
Gs)
in
th
at
the
form
er
per
form
de
term
inis
tic o
per
atio
ns o
n a
see
d va
lue,
whi
le t
he l
att
er r
ely
on a
n
on-d
eter
min
istic
so
urce
of
rand
om
nes
s (u
sua
lly a
phy
sica
l pr
oces
s su
ch
as
radi
oact
ive
deca
y or
th
erm
al
noi
se).
Tw
o (o
f m
any
) pr
ope
rtie
s th
at
a “
goo
d” P
RN
G o
r R
NG
sho
uld
have
are
: (
1)
the
num
ber
s of
bits
‘0’
and
‘1
’ it
prod
uces
sh
ould
be
alm
ost
eq
ual,
and
(2
) th
e se
que
nce
sho
uld
not
rep
eat,
tha
t is,
it s
houl
d b
e n
on-
peri
odic
.
In a
n ea
rlie
r w
ork
(Noc
he,
20
04)
, a
cer
tain
bin
ary
se
que
nce
ge
nera
tor
wa
s pr
opos
ed.
It
star
ted
with
a T
hu
e-M
orse
seq
uenc
e 0
, 1
, 1
, 0
, 1
, 0
, 0,
1,
1, 0
, 0
, 1,
0,
1, 1
, 0
, …
(se
que
nce
A01
00
60
in S
loa
ne
(2
006
)).
Thi
s se
que
nce
is n
ot p
erio
dic
(Allo
uch
e &
Sh
alli
t, 1
99
9) a
nd
the
nu
mb
ers
of ‘
0’s
and
‘1
’s i
n a
n in
finite
Th
ue-M
orse
se
que
nce
are
th
e sa
me.
T
his
seq
uenc
e, h
ow
eve
r, t
erri
bly
fa
ils c
om
mo
nly
acc
ept
ed
st
atis
tica
l tes
ts o
f ra
ndo
mn
ess.
It w
as s
how
n in
(N
och
e, 2
00
4) th
at
the
alg
orith
m u
sed
in t
hat
wor
k to
ge
nera
te t
he T
hue
-Mor
se s
eque
nce
lea
d to
an
int
eres
ting
KA
MA
WO
TA
N v
ol.
1,
no. 2
, M
ay
20
07
80
met
hod
of
ad
din
g “d
isor
der”
to
the
orig
ina
l seq
ue
nce
. T
he b
its w
hos
e
orde
r in
th
e se
qu
enc
e w
as
a m
ultip
le o
f a
pri
me
nu
mb
er p
wer
e in
vert
ed,
an
d th
is w
as d
one
seq
uen
tially
for
incr
easi
ng
p fo
r th
e fir
st f
odd
pri
me
nu
mb
ers.
(O
dd p
rim
e n
um
ber
s w
ere
chos
en b
eca
use
no
“dis
orde
r” i
s in
trod
uce
d b
y in
vert
ing
all
the
eve
n b
its,
tha
t is
, th
e m
ultip
les
of t
he o
nly
eve
n pr
ime
nu
mb
er,
2.)
Thi
s ge
nera
tor
is
now
m
odifi
ed
by
sta
rtin
g it
with
a
K
ola
kosk
i seq
uenc
e in
stea
d.
A K
ola
kosk
i seq
uen
ce is
a
seq
uenc
e th
at
equa
ls it
s ow
n ru
n le
ngt
h se
que
nce
(K
ola
kosk
i, 1
96
5).
T
he s
eque
nce
()
{}∞ =0n
nk
= 1
, 2,
2,
1, 1
, 2,
1,
2, 2
, 1,
2,
2, 1
, 1,
2,
… (
sequ
en
ce
A0
00
002
in S
loa
ne (
20
06
)) is
the
Kol
ako
ski s
equ
enc
e t
hat
star
ts w
ith
‘1.’
Lis
ted
abo
ve a
re it
s fir
st t
en r
uns;
the
fir
st r
un is
‘1,’
the
seco
nd
run
is ‘2
,2,’
the
thir
d ru
n is
‘1,1
,’ th
e fo
urth
ru
n is
‘2,’
and
so
on.
T
he
fir
st r
un h
as k
(1)
= 1
ele
me
nt,
the
seco
nd
run
has
k(2)
= 2
ele
men
ts,
the
thir
d ru
n ha
s k(3
) =
2 e
lem
ent
s, t
he f
ourt
h ru
n h
as k(4)
= 1
el
em
ent
, a
nd s
o o
n.
The
Kol
ako
ski
seq
uenc
e th
at
star
ts w
ith ‘2
’ is
2,
2, 1
, 1
, 2
, 1
, 2
, 2
, 1
, 2
, 2,
1,
1, 2
, 1,
1,
… (
seq
uenc
e A
07
888
0 i
n S
loa
ne
(20
06))
.
Thi
s is
the
ver
sio
n a
rbitr
arily
cho
sen
in t
his
wor
k.
A b
ina
ry s
eque
nce
co
mp
ose
d of
‘0
’s
and
‘1
’s
is
desi
red,
so
the
‘2’
s a
re a
rbi
trar
ily
conv
erte
d in
to ‘0
’s t
o g
et th
e b
ina
ry s
eque
nce (
){
}∞ =0nn
x =
0,
0,
1, 1
, 0,
1,
0, 0
, 1
, 0,
0,
1, 1
, 0,
1,
1, …
.
The
K
ola
kosk
i se
que
nce
is
not
peri
odic
(K
ola
kosk
i a
nd
Uco
luk,
19
66
) a
nd i
t is
co
njec
ture
d (b
ut n
ot y
et p
ro
ven)
tha
t th
e
num
ber
of
‘0’s
and
‘1
’s i
n an
inf
inite
Kol
ako
ski
seq
uen
ce a
re t
he
sa
me
(Wei
sste
in, 2
00
6).
Met
hodo
logy
One
pop
ula
r w
ay
to t
est
PR
NG
s a
nd R
NG
s is
the
DIE
HA
RD
se
ries
of t
ests
(M
ars
agl
ia,
199
5).
DIE
HA
RD
co
ntai
ns fi
fte
en
test
s fo
r ra
ndo
mn
ess
of u
nifo
rmly
dis
trib
uted
nu
mb
ers.
C
are
sho
uld
be
take
n
whe
n ru
nni
ng
thes
e te
sts.
F
or e
xam
ple,
Mar
sagl
ia r
epor
ts t
hat
the
hard
wa
re
RN
Gs
he
test
ed
faile
d “s
pect
acu
larl
y.”
D
avi
es
(19
97)
fo
un
d th
at
the
sup
pose
d fa
ilure
s w
ere
mos
tly d
ue
to a
da
ta h
an
dlin
g
erro
r:
each
byt
e w
ith a
va
lue
of 1
0 w
as
bei
ng r
ecor
de
d a
s tw
o b
ytes
w
ith v
alu
es o
f 13
and
10
(a
car
ria
ge r
etur
n fo
llow
ed b
y a
lin
e fe
ed)
.
An
othe
r po
ssib
le p
rob
lem
is t
he d
iffer
ent
con
vent
ions
fo
r th
e or
der
in
whi
ch t
he
bits
or
byt
es a
re s
tore
d in
a w
ord.
T
he
rea
rra
nge
me
nt o
f b
ytes
whe
n c
op
yin
g b
etw
een
com
put
ers
with
diff
eren
t c
on
vent
ions
Col
lege
of
Art
s a
nd
Sci
en
ces,
Ate
neo
de
Na
ga
Un
ive
rsi
ty
81
can
affe
ct t
he r
esul
ts o
f co
rrel
atio
ns t
ests
an
d ot
her
test
s (D
avie
s,
200
0).
In
thi
s w
ork,
bot
h pr
oble
ms
are
avo
ide
d:
the
out
put
is
save
d
as
a b
inar
y fil
e (n
ot a
tex
t fil
e),
and
onl
y b
yte-
siz
ed w
ord
s ar
e us
ed
to
repr
esen
t th
e se
qu
ence
. D
eta
ils o
f th
e te
sts
an
d th
e
ass
ocia
ted p
-va
lues
are
in
Ma
rsa
glia
(1
99
5).
The
res
ults
of
the
test
s ca
n b
e
sum
ma
rize
d b
y 1
46
p-
valu
es.
The
pro
gra
m p
rese
nte
d in
(N
och
e, 2
00
4)
calle
d su
bro
utin
es
fro
m in
side
loo
ps.
Th
e re
vise
d ve
rsio
n lis
ted
in A
ppe
ndi
x A
doe
s n
ot
use
sub
rout
ine
calls
, res
ultin
g in
impr
ove
d p
erfo
rma
nce
.
Fo
ur c
ase
s a
re t
este
d:
thos
e in
volv
ing
the
first
f
= 1
0,
10
0,
100
0,
and
20
00
odd
pri
me
nu
mb
ers.
T
he
case
s la
bele
d t
10
, t1
00
, t1
00
0, a
nd t
20
00 u
se t
he T
hu
e-M
orse
seq
uenc
e, a
nd
tho
se l
ab
eled
k1
0,
k10
0,
k10
00
, a
nd k
20
00
use
the
Kol
ako
ski s
eque
nc
e.
(The
latt
er
four
ca
ses
use
the
pro
gra
m l
iste
d in
Ap
pen
dix
B.)
E
ach
seq
ue
nce
is
16,
77
7,2
16
byte
s lo
ng.
(D
IEH
AR
D n
eeds
at
lea
st a
n 8
0 m
illio
n b
it lo
ng te
st s
eque
nce
for
all
of it
s te
sts
to r
un.)
For
co
mpa
riso
n, t
he r
esul
ts o
f th
e P
RN
G K
ISS
(M
arsa
gli
a,
199
5)
are
also
inc
lud
ed.
It
“se
ems
to p
ass
all
[the
DIE
HA
RD
] te
sts
and
is
hig
hly
reco
mm
en
ded
for
sp
eed
and
sim
plic
ity”
(Mar
sagl
ia,
199
5).
T
he
KIS
S s
equ
enc
e te
ste
d h
ere
is 1
1,4
68,
80
0 b
yte
s lo
ng a
nd
use
d th
e se
ed
inte
gers
1,
4, 3
, a
nd 4
4.
Res
ults
Ta
ble
1 s
how
s th
e ru
n tim
es f
or a
co
mp
uter
with
an
Int
el
Cel
eron
1.7
0 G
Hz
proc
esso
r, 2
56
MB
of
RA
M,
and
a M
icr
osof
t W
ind
ows
XP
ope
ratin
g sy
ste
m.
The
KIS
S o
utpu
t fil
e ta
kes
less
tha
n
a s
econ
d to
ge
nera
te a
fter
the
seed
va
lues
are
giv
en.
Ta
ble
1. R
un ti
mes
(in
sec
onds
)
f T
hue-
Mor
se
(Noc
he,
20
04)
T
hue
-Mor
se
(App
endi
x A
) K
ola
kosk
i (A
ppen
dix
B)
10
1
3
4 8
1
00
18
7
11
1
00
0 2
6
15
1
8
20
00
29
1
6
20
The
p-v
alu
es f
or t
he t
10
, t1
00,
t1
000
, t2
00
0,
and
KIS
S t
est
ca
ses
are
lis
ted
in (
Noc
he,
20
04
).
(Th
e pr
ogr
am
in
Ap
pen
dix
A
yiel
ds t
he s
am
e D
IEH
AR
D
p-va
lues
as
the
pro
gra
m in
Noc
he
(20
04).
)
KA
MA
WO
TA
N v
ol.
1,
no. 2
, M
ay
20
07
82
Due
to
spa
ce l
imita
tions
, th
e p-va
lues
for
the
k1
0,
k10
0,
k10
00
, an
d
k20
00
ca
ses
are
not
liste
d he
re.
Dis
cuss
ion
It i
s im
plie
d t
hat
if a
ll th
e p-
valu
es o
f a
seq
ue
nce
tes
ted
by
DIE
HA
RD
lie
in t
he r
ang
e (0
.025
, 0.
97
5) t
hen
the
seq
uen
ce is
sa
id t
o ha
ve p
ass
ed a
ll th
e D
IEH
AR
D t
ests
. T
his
is t
he r
easo
ning
be
hin
d th
e
cla
im in
(N
oche
, 20
04
) th
at
the
seq
uen
ce t
20
00
per
form
ed
wel
l in
the
te
sts.
A
sub
seq
uent
ana
lysi
s of
the
da
ta,
how
eve
r, s
how
s th
at
t20
00’
s te
st r
esul
ts s
till l
eave
mor
e to
be
desi
red.
The
his
togr
am
s of
the
p-va
lues
for
ea
ch T
hue
-Mor
se-b
ased
te
st c
ase
are
sho
wn
in F
igur
e 1
. N
ote
tha
t m
ost
of t
he p
-va
lues
are
in
the
upp
erm
ost
deci
le.
The
his
togr
am
for
the
KIS
S t
est
ca
se (
Fig
ure
2)
is m
ore
uni
form
ly d
istr
ibut
ed.
Thi
s se
em
s to
im
ply
tha
t a
“m
ore
ra
ndo
m”
seq
ue
nce
wo
uld
have
a m
ore
uni
form
ly d
istr
ibu
ted
set
of p-
valu
es in
the
DIE
HA
RD
test
s.
Fig
ure
1.
His
togr
am
s of
DIE
HA
RD
p-va
lues
(T
hue-
Mor
se-b
ase
d se
quen
ces)
0.0-
0.1
0.1
-0.2
0.2-
0.3
0.3-
0.4
0.4
-0.5
0.5
-0.6
0.6
-0.7
0.7
-0.8
0.8-
0.9
0.9
-1.0
t200
0
t100
0
t100t1
00
20406080
100
120
140
Col
lege
of
Art
s a
nd
Sci
en
ces,
Ate
neo
de
Na
ga
Un
ive
rsi
ty
83
Fig
ure
2.
His
togr
am
of D
IEH
AR
D p-
valu
es (
KIS
S s
eque
nce
)
0.0-
0.1
0.1-
0.2
0.2-
0.3
0.3-
0.4
0.4-
0.5
0.5-
0.6
0.6-
0.7
0.7-
0.8
0.8-
0.9
0.9-
1.0
KIS
S
0
20406080
100
120
140
Fig
ure
3.
His
togr
am
s of
DIE
HA
RD
p-va
lues
(K
ola
kosk
i-ba
sed
sequ
ence
s)
0.0-
0.1
0.1-
0.2
0.2
-0.3
0.3
-0.4
0.4-
0.5
0.5
-0.6
0.6-
0.7
0.7
-0.8
0.8-
0.9
0.9
-1.0
k200
0
k100
0
k10
0
k10
0
20406080
100
120
140
KA
MA
WO
TA
N v
ol.
1,
no. 2
, M
ay
20
07
84
Fig
ure
3 s
how
s th
e hi
stog
ram
s fo
r th
e K
ola
kosk
i-ba
sed
test
ca
ses.
It
show
s th
at e
ven
case
k1
00
see
ms
to h
ave
be
tter
te
st r
esul
ts
tha
n ca
se t
20
00.
It m
ust
be
me
ntio
ned,
how
eve
r, t
hat
all
the
Kol
ako
ski
-bas
ed
test
ca
ses
perf
orm
ed
very
p
oor
ly
in
the
se
vent
h D
IEH
AR
D
test
(“
Cou
nt t
he 1
’s in
a s
trea
m o
f b
ytes
”).
Thi
s ca
n b
e s
een
in T
ab
le 2
.
Ta
ble
2. p
-va
lue
s fo
r th
e se
vent
h D
IEH
AR
D te
st (
roun
ded
off)
.
t1
0 t1
00
t100
0 t2
000
K
ISS
k1
0 k1
00
k100
0 k2
000
p 109
1
.000
0.
979
0.96
3 0.
997
0.9
19
1.00
0 1.
000
1.00
0 0.
993
p 110
1
.000
1.
000
1.00
0 0.
835
0.3
36
1.00
0 1.
000
1.00
0 1.
000
Con
clus
ions
The
bin
ary
seq
uenc
e ge
ner
ato
r pr
esen
ted
her
e is
sh
own
to
pe
rfor
m a
rgua
bly
bet
ter
in a
pop
ular
set
of
test
s fo
r r
and
om
nes
s th
an
a
n ea
rlie
r w
ork.
In
pa
rtic
ula
r, c
ase
k1
00 h
as
a s
hor
ter
run
tim
e th
an
ca
se t
20
00
and
its
p-
valu
es a
re m
ore
uni
form
ly d
istr
ibut
ed
tha
n th
ose
of
ca
se t
20
00.
Ref
eren
ces
Allo
uche
, J.
-P.,
& S
halli
t, J.
(19
99).
The
ubi
quitous
Pro
uhet
-Thu
e-M
orse
se
quen
ce.
In Se
que
nce
s a
nd T
heir
A
ppl
ica
tions
: P
roce
edin
gs of
S
ET
A ’9
8, S
prin
ger-
Ver
lag,
1–1
6.
Da
vies
, R
. (1
997)
. Tru
e r
an
dom
num
ber
ge
nera
tors. R
etri
eved
Apr
il 20
04,
from
http
://w
ww
.rob
ertn
z. n
et/tr
ue_m
g.ht
ml
Da
vies
, R
. (2
000)
. Ha
rdw
are
ran
dom
num
ber
gen
erat
ors
. R
etrie
ved
Apr
il 20
04, f
rom
http
://w
ww
.rob
ertn
z.ne
t/hw
rng.
htm
l
Ha
yes,
B.
(200
1).
Com
putin
g sc
ienc
e: R
ando
mne
ss a
s a
res
ourc
e, A
mer
ica
n S
cie
ntis
t, 89(
4), 3
00–3
04.
Kol
ako
ski,
W.
(196
5).
Sel
f ge
nera
ting
runs
, P
robl
em 53
04.
Am
eric
an
Ma
the
ma
tical
Mo
nth
ly, 7
2(6)
, 674
.
Kol
ako
ski,
W.,
&
Uco
luk,
N
. (1
966)
. 53
04. A
me
rican
M
ath
em
atic
al
Mo
nth
ly, 7
3(6)
, 68
1–68
2.
Mar
sagl
ia,
G.
(199
5). The
Mar
sag
lia r
an
dom
num
ber
CD
RO
M in
clu
din
g t
he
DIE
HA
RD
bat
tery
of
test
s of
ra
ndom
ness
. R
etrie
ved
Ma
rch
2004
, fr
om h
ttp://
sta
t.fsu
.edu
/pub
/die
hard
/
Noc
he,
J. (
2004
). A
bin
ary
seq
uenc
e ge
nera
tor
base
d o
n th
e T
hue-
Mor
se
sequ
ence
a
nd
mul
tiple
s of
pr
imes
. T
he
Ph
ilipp
ine
S
tatis
ticia
n, 5
3(1–
4),
67–7
6.
Col
lege
of
Art
s a
nd
Sci
en
ces,
Ate
neo
de
Na
ga
Un
ive
rsi
ty
85
Slo
ane
, N
. (2
006)
. Th
e o
n-li
ne e
ncyc
lope
dia
of
inte
ger
sequ
en
ces
. R
etri
eved
A
pril
2007
, fr
om h
ttp:
//ww
w.r
esea
rch.
att.c
om/~
njas
/se
quen
ces/
Wei
sste
in,
E.
(200
6). K
olak
oski
se
que
nce. F
rom
Ma
thW
orld
—A
Wo
lfra
m
We
b
Re
sour
ce. R
etrie
ved
Apr
il 20
07
from
ht
tp://
ma
thw
orld
.w
olfr
am
.com
/Kol
ako
skiS
eque
nce.
htm
l
App
endi
x A
/* improved version of the program presented in (Noche, 2004) */
/* source code for Microsoft Visual C++ 6.0 */
#include <stdio.h>
#include <stdlib.h>
#define M 27
#define L 134217728
#define B 16777216
#define F 2000
#define MAXPRIME 17393
/* F=10, MAXPRIME=31
F=100, MAXPRIME=547
F=1000, MAXPRIME=7927
F=2000, MAXPRIME=17393 */
void main(void)
{
unsigned char bitconst[8] =
{0x80, 0x40, 0x20, 0x10, 0x08, 0x04, 0x02, 0x01};
unsigned long j = 0, k;
unsigned short i;
unsigned char *x;
FILE *fp;
unsigned short *prime;
prime = (unsigned short *) malloc((F+1) * sizeof(unsigned
short));
unsigned char *c;
c = (unsigned char *) malloc((MAXPRIME+1) * sizeof(unsigned
char));
for (i = 1; i <= MAXPRIME; i++)
c[i] = 1;
c[1] = 0;
prime[0] = 2;
do
{
for (i = prime[j]; i <= MAXPRIME; i += prime[j])
c[i] = 0;
i = prime[j] + 1;
while (c[i] == 0)
i++;
prime[++j] = i;
}
while (j < F);
free(c);
KA
MA
WO
TA
N v
ol.
1,
no. 2
, M
ay
20
07
86
x = (unsigned char *) malloc(B * sizeof(unsigned char));
x[0] = 0x69;
x[1] = ~x[0];
for (i = 1; i < (M-3); i++)
{
k = 1 << i; /* k = 2^i */
for (j = 0; j < k; j++)
x[k+j] = ~x[j];
}
for (i = 1; i <= F; i++)
for (j = prime[i]-1; j < L; j += prime[i])
x[j/8] ^= bitconst[j%8];
fp = fopen("t.dat", "wb");
fwrite(x, B, 1, fp);
fclose(fp);
}
App
endi
x B
/* source code for Microsoft Visual C++ 6.0 */
#include <stdio.h>
#include <stdlib.h>
#define M 27
#define L 134217728
#define B 16777216
#define F 2000
#define MAXPRIME 17393
/* F=10, MAXPRIME=31
F=100, MAXPRIME=547
F=1000, MAXPRIME=7927
F=2000, MAXPRIME=17393
L is the length of the sequence in bits; L = 2^M
B is the size of the sequence in bytes
F is the number of (odd) primes to use
MAXPRIME is the Fth odd prime (the (F+1)th prime) */
void main(void)
{
unsigned char bitconst[8] =
{0x80, 0x40, 0x20, 0x10, 0x08, 0x04, 0x02, 0x01};
unsigned long j = 0, k;
unsigned short i;
unsigned char *x;
FILE *fp;
/* prime[0] = 2; prime[1] = 3; the Fth odd prime is prime[F] */
unsigned short *prime;
prime = (unsigned short *) malloc((F+1) * sizeof(unsigned
short));
unsigned char *c;
Col
lege
of
Art
s a
nd
Sci
en
ces,
Ate
neo
de
Na
ga
Un
ive
rsi
ty
87
c = (unsigned char *) malloc((MAXPRIME+1) * sizeof(unsigned
char));
/* data types might need to be changed if F is increased */
for (i = 1; i <= MAXPRIME; i++)
c[i] = 1;
c[1] = 0;
prime[0] = 2;
do
{
for (i = prime[j]; i <= MAXPRIME; i += prime[j])
c[i] = 0;
i = prime[j] + 1;
while (c[i] == 0)
i++;
prime[++j] = i;
}
while (j < F);
free(c);
/* create Kolakoski word */
x = (unsigned char *) malloc(B * sizeof(unsigned char));
x[0] = 0x34;
for (j = 1; j < B; j++)
x[j] = 0x00;
k = 8;
for (j = 5; k < L; j++)
{
x[k/8] |= bitconst[k%8];
k++;
if ((x[j/8] & bitconst[j%8]) == 0x00)
{
x[k/8] |= bitconst[k%8];
k++;
}
j++;
k++;
if ((x[j/8] & bitconst[j%8]) == 0x00)
k++;
}
/* invert bits that are multiples of the first F odd primes */
for (i = 1; i <= F; i++)
for (j = prime[i]-1; j < L; j += prime[i])
x[j/8] ^= bitconst[j%8];
fp = fopen("k.dat", "wb");
fwrite(x, B, 1, fp);
fclose(fp);
}