A118255 -id:A118255 - cp's OEIS frontend

A118256 Concatenation for i=1 to n of A005171(i); also A118255 in base 2.

1, 10, 100, 1001, 10010, 100101, 1001010, 10010101, 100101011, 1001010111, 10010101110, 100101011101, 1001010111010, 10010101110101, 100101011101011, 1001010111010111, 10010101110101110, 100101011101011101, 1001010111010111010, 10010101110101110101, 100101011101011101011

Offset: 1

Pierre CAMI, Apr 19 2006

			A005171 : 1,0,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1 ................
a(1)=1, a(2)=10, a(3)=100, a(4)=1001, ...

Michael S. Branicky, Table of n, a(n) for n = 1..1000 (terms 1..30 from N. J. A. Sloane)

Cf. A005171, A118255, A118257.

Array[FromDigits@ Array[Boole[! PrimeQ@ #] &, #] &, 21] (* or *)
FromDigits@ IntegerDigits[#, 2] & /@ Last@ Transpose@ NestList[{#1 + 1, If[PrimeQ[#1 + 1], 2 #2, 2 #2 + 1]} & @@ # &, {1, 1}, 21] (* Michael De Vlieger, Nov 01 2016, latter after Harvey P. Dale at A118255 *)

a(n) = sum(k=1, n, !isprime(k)*10^(n-k)); \\ Michel Marcus, Nov 01 2016

from sympy import isprime
def a(n): return int("".join(str(1-isprime(i)) for i in range(1, n+1)))
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Jan 10 2022

# faster version for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    an = 0
    for k in count(1):
        an = 10 * an + int(not isprime(k))
        yield an
print(list(islice(agen(), 21))) # Michael S. Branicky, Jan 10 2022

a(n) ~ 10^n * 0.10010101.... [Charles R Greathouse IV, Dec 27 2011]

Corrected by Omar E. Pol, Nov 08 2007

A118257 Numbers k such that A118255(k) is prime.

Original entry on oeis.org

2, 6, 8, 10, 18, 26, 30, 74, 142, 203, 398, 651, 792, 1314, 3487, 5978, 6240, 7814, 8054, 8673, 21436, 23947, 52985, 91784, 157537, 164901

Offset: 1

Pierre CAMI, Apr 19 2006

A118255(1314) is prime with 396 digits

A118255(23947) is a probable prime with 7209 digits. - Giovanni Resta, Apr 26 2006

			A118255(2) = 2 prime, A118255(6) = 149 prime, A118255(8) = 599 prime.

Cf. A005171, A118255, A118256.

A118255[n_] := Module[{},
   If[n == 1, A118255[1] = 1,
    If[PrimeQ[n], A118255[n] = 2 A118255[n - 1],
     A118255[n] = 2 A118255[n - 1] + 1]]];
Select[Range[5000], PrimeQ[A118255[#]] &] (* Robert Price, Apr 03 2019 *)

a(15)-a(22) from Giovanni Resta, Apr 26 2006

a(23)-a(26) from Michael S. Branicky, Dec 11 2024

A139102 Numbers whose binary representation shows the distribution of prime numbers up to the n-th prime minus 1, using "0" for primes and "1" for nonprime numbers.

Original entry on oeis.org

1, 2, 9, 37, 599, 2397, 38359, 153437, 2454999, 157119967, 628479869, 40222711647, 643563386359, 2574253545437, 41188056726999, 2636035630527967, 168706280353789919, 674825121415159677, 43188807770570219359, 691020924329123509751, 2764083697316494039005

Offset: 1

Omar E. Pol, Apr 08 2008

a(n) is the decimal representation of A139101(n) interpreted as binary number.

			a(4)=37 because 37 written in base 2 is 100101 and the string "100101" shows the distribution of prime numbers up to the 4th prime minus 1, using "0" for primes and "1" for nonprime numbers.

Michael S. Branicky, Table of n, a(n) for n = 1..468
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Subset of A118255.

Cf. A000040, A018252, A139101, A139103, A139104, A139119, A139120, A139122.

A139101 := proc(n) option remember ; local a,p; if n = 1 then RETURN(1); else a := 10*A139101(n-1) ; for p from ithprime(n-1)+1 to ithprime(n)-1 do a := 10*a+1 ; od: fi ; RETURN(a) ; end: # R. J. Mathar, Apr 25 2008
bin2dec := proc(n) local nshft ; nshft := convert(n,base,10) ; add(op(i,nshft)*2^(i-1),i=1..nops(nshft) ) ; end: # R. J. Mathar, Apr 25 2008
A139102 := proc(n) bin2dec(A139101(n)) ; end: # R. J. Mathar, Apr 25 2008
seq(A139102(n),n=1..35) ; # R. J. Mathar, Apr 25 2008

Table[ sum = 0; For[i = 1, i <= Prime[n] - 1 , i++, sum = sum*2;
If[! PrimeQ[i], sum++]]; sum, {n, 1, 25}] (* Robert Price, Apr 03 2019 *)

a(n) = fromdigits(vector(prime(n)-1, k, !isprime(k)), 2); \\ Michel Marcus, Apr 04 2019

from sympy import isprime, prime
def a(n):
    return int("".join(str(1-isprime(i)) for i in range(1, prime(n))), 2)
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Jan 10 2022

# faster version for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    an = 0
    for k in count(1):
        an = 2 * an + int(not isprime(k))
        if isprime(k+1):
            yield an
print(list(islice(agen(), 21))) # Michael S. Branicky, Jan 10 2022

a(n) = A139104(n)/2.

More terms from R. J. Mathar, Apr 25 2008

a(20)-a(21) from Robert Price, Apr 03 2019

A139104 Numbers whose binary representation shows the distribution of prime numbers up to the n-th prime, using "0" for primes and "1" for nonprime numbers.

Original entry on oeis.org

2, 4, 18, 74, 1198, 4794, 76718, 306874, 4909998, 314239934, 1256959738, 80445423294, 1287126772718, 5148507090874, 82376113453998, 5272071261055934, 337412560707579838, 1349650242830319354, 86377615541140438718, 1382041848658247019502, 5528167394632988078010

Offset: 1

Omar E. Pol, Apr 08 2008

nonn

a(n) is the decimal representation of A139103(n) interpreted as binary number.

			a(4)=74 because 74 written in base 2 is 1001010 and the string "1001010" shows the distribution of prime numbers up to the 4th prime, using "0" for primes and "1" for nonprime numbers.

Harvey P. Dale, Table of n, a(n) for n = 1..467
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Subset of A118255.

Cf. A000040, A018252, A139101, A139102, A139103, A139119, A139120, A139122, A000720, A001348, A121240.

Table[ sum = 0; For[i = 1, i <= Prime[n] , i++, sum = sum*2;
If[! PrimeQ[i], sum++]]; sum, {n, 1, 21}] (* Robert Price, Apr 03 2019 *)
Module[{nn=30,t},t=Table[If[PrimeQ[n],0,1],{n,Prime[nn]}];Table[ FromDigits[ Take[t,p],2],{p,Prime[Range[nn]]}]] (* Harvey P. Dale, Jul 15 2019 *)

a(n) = fromdigits(vector(prime(n), k, !isprime(k)), 2); \\ Michel Marcus, Apr 04 2019

a(n) = 2 * A139102(n).
From Ridouane Oudra, Aug 27 2019: (Start)
a(n) = 2^prime(n) - 1 - (1/2)*(n + Sum_{i=1..prime(n)} 2^(prime(n)-i)*pi(i)), where prime(n) = A000040(n) and pi(n) = A000720(n)
a(n) = A001348(n) - A121240(n)
a(n) = A118255(A000040(n)). (End)

More terms from R. J. Mathar, May 22 2008

a(19)-a(21) from Robert Price, Apr 03 2019

A072762 n coded as binary word of length=n with k-th bit set iff k is prime (1<=k<=n), decimal value.

Original entry on oeis.org

0, 1, 3, 6, 13, 26, 53, 106, 212, 424, 849, 1698, 3397, 6794, 13588, 27176, 54353, 108706, 217413, 434826, 869652, 1739304, 3478609, 6957218, 13914436, 27828872, 55657744, 111315488, 222630977, 445261954, 890523909, 1781047818, 3562095636, 7124191272

Offset: 1

Reinhard Zumkeller, Aug 08 2002

a(n) is odd iff n is prime.
a(p) where p is prime is the numerator of Sum_{q <= p} 1/2^q where the sum is over primes up to p. - Alexander Adamchuk, Aug 22 2006
The n-th approximation to the Prime Constant is given by a(n)/2^n. - Anton Vrba (antonvrba(AT)yahoo.com), Nov 24 2006

			a(6) = '011010' = (((0*2+1)*2+1)*2*2+1)*2 = 26.
a(7) = '0110101' = (((0*2+1)*2+1)*2*2+1)*2*2+1 = 53.
a(8) = '01101010' = ((((0*2+1)*2+1)*2*2+1)*2*2+1)*2 = 106.

Alois P. Heinz, Table of n, a(n) for n = 1..3323 (first 300 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A051006, A100634, A010051, A000720, A118255.

a072762 n = foldl (\v d -> 2*v + d) 0 $ map a010051 [1..n]
-- Reinhard Zumkeller, Sep 17 2011

a:= proc(n) option remember;
      `if`(n<2, 0, 2 * a(n-1) + `if`(isprime(n), 1, 0))
    end:
seq(a(n), n=1..40);  #  Alois P. Heinz, Jan 18 2011

a[1] = 0; a[n_] := a[n] = 2*a[n-1] + Boole[PrimeQ[n]]; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Jun 14 2013 *)
nxt[{n_,a_}]:={n+1,Boole[PrimeQ[n+1]]+2a}; Transpose[NestList[nxt,{1,0},30]][[2]] (* Harvey P. Dale, Jan 07 2015 *)

an=0; print1(an,", "); for(n=2,31, an=2*an+isprime(n); print1(an,", ")) \\ Washington Bomfim, Jan 18 2011

a(n)=my(s=1,p=2);forprime(q=3,n,s=s<<(q-p)+1;p=q);s<<(n-p) \\ Charles R Greathouse IV, Jun 03 2013

a(1) = 0 and a(n) = a(n-1)*2 + A010051(n) for n>1.
a(n) = (1/2)*(pi(n) + Sum_{i=1..n} 2^(n-i)*pi(i)), where pi = A000720. - Ridouane Oudra, Aug 26 2019
a(n) = floor(c*2^n), where c = A051006 is the prime constant. - Lorenzo Sauras Altuzarra, Jan 03 2023

A139119 Primes whose binary representation shows the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.

Original entry on oeis.org

2, 37, 149, 599, 153437, 39279991, 628479869, 11056334789265976156021, 3263254052013454238294691704608897001027543, 7524551543123483484068003542235060639999919940760883731360687

Offset: 1

Omar E. Pol, Apr 11 2008

Primes in A118255.
Primes whose binary representation is also the concatenation of the initial terms of A005171, the characteristic function of nonprimes. - Omar E. Pol, Oct 07 2013
a(11) is a 120-digit number 377859...798653. - Robert Price, Apr 03 2019

Cf. A000040, A118255, A118256, A118257, A139101, A139102, A139103, A139104.

Select[Table[FromDigits[Boole /@ Not /@ PrimeQ /@ Range@k, 2], {k, 1, 100}], PrimeQ] (* Federico Provvedi, Oct 07 2013 *)

f(n) = fromdigits(vector(n, k, !isprime(k)), 2); \\ A118255
lista(nn) = for (n=1, nn, if (isprime(p=f(n)), print1(p, ", "))); \\ Michel Marcus, Apr 04 2019

a(8)-a(10) from Donovan Johnson, Oct 07 2013

A139120 Primes that show the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.

Original entry on oeis.org

10010101, 1001010111010111, 100101011101011101011101111101011111011101011101111101111101011111011101011111011101111101111111011101011101011101111111111111011101111101011111111101011111011111011101111101111, 100101011101011101011101111101011111011101011101111101111101011111011101011111011101111101111111011101011101011101111111111111011101111101011111111101011111011111011101111101111101011111111101011101011111111

Offset: 1

Omar E. Pol, Apr 11 2008

Primes in A118256.

For n = 1..7, the number of digits in a(n) is 8, 16, 177, 207, 872, 1395, 2114 (no more through 10000). - Jon E. Schoenfield, Apr 13 2018

Alois P. Heinz, Table of n, a(n) for n = 1..5
Caldwell and Honaker, Prime Curios!: 10010101

Cf. A118255, A118256, A118257, A139101, A139102, A139103, A139104, A139119, A139122.

A118255[n_] := Module[{},
   If[n == 1, A118255[1] = 1,
    If[PrimeQ[n], A118255[n] = 2 A118255[n - 1],
     A118255[n] = 2 A118255[n - 1] + 1]]];
Select[Table[FromDigits[IntegerDigits[A118255[n], 2]], {n, 1, 1000}], PrimeQ] (* Robert Price, Apr 03 2019 *)

Extended by Charles R Greathouse IV, Jul 27 2009

A139122 Primes whose binary representation shows the distribution of prime numbers up to some prime minus 1, using "0" for primes and "1" for nonprime numbers.

Original entry on oeis.org

2, 37, 599, 153437, 628479869

Offset: 1

Omar E. Pol, Apr 11 2008

Primes in A139102.

a(6) > 10^14632 if it exists (no further primes in first 5000 terms of A139102). - Michael S. Branicky, Jan 25 2022

Cf. A118255, A118256, A118257, A139101, A139102, A139103, A139104, A139119, A139120.

Select[Table[ sum = 0; For[i = 1, i <= Prime[n] - 1 , i++, sum = sum*2; If[! PrimeQ[i], sum++]]; sum, {n, 1, 1000}], PrimeQ[#] &] (* Robert Price, Apr 03 2019 *)
Module[{nn=500,p,x},p=Table[If[PrimeQ[n],0,1],{n,nn}];x=SequencePosition[p,{1,0}][[All,1]];Join[{2},Select[Table[FromDigits[Take[p,k],2],{k,x}],PrimeQ]]] (* Harvey P. Dale, Jun 15 2022 *)

f(n) = fromdigits(vector(prime(n)-1, k, !isprime(k)), 2); \\ A139102
lista(nn) = for (n=1, nn, if (isprime(p=f(n)), print1(p, ", ")));

# uses agen() in A139102
from sympy import isprime
print(list(islice(filter(isprime, agen()), 5))) # Michael S. Branicky, Jan 25 2022

a(5) from Robert Price, Apr 03 2019

cp's OEIS Frontend

A118256 Concatenation for i=1 to n of A005171(i); also A118255 in base 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Formula

Extensions

A118257 Numbers k such that A118255(k) is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A139102 Numbers whose binary representation shows the distribution of prime numbers up to the n-th prime minus 1, using "0" for primes and "1" for nonprime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A139104 Numbers whose binary representation shows the distribution of prime numbers up to the n-th prime, using "0" for primes and "1" for nonprime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A072762 n coded as binary word of length=n with k-th bit set iff k is prime (1<=k<=n), decimal value.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

A139119 Primes whose binary representation shows the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A139120 Primes that show the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.