A118256 -id:A118256 - cp's OEIS frontend

A139101 Numbers that show the distribution of prime numbers up to the n-th prime minus 1, using "0" for primes and "1" for nonprime numbers.

1, 10, 1001, 100101, 1001010111, 100101011101, 1001010111010111, 100101011101011101, 1001010111010111010111, 1001010111010111010111011111, 100101011101011101011101111101, 100101011101011101011101111101011111, 1001010111010111010111011111010111110111

Offset: 1

Omar E. Pol, Apr 08 2008

a(n) has A000040(n)-1 digits, n-1 digits "0" and A000040(n)-n digits "1".

Michael S. Branicky, Table of n, a(n) for n = 1..168
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Binary representation of A139102.
Subset of A118256.
Cf. A000040, A018252, A139103, A139104, A139119, A139120, A139122.

Table[ sum = 0; For[i = 1, i <= Prime[n] - 1 , i++, sum = sum*2;
If[! PrimeQ[i], sum++]]; IntegerString[sum, 2], {n, 1, 13}] (* Robert Price, Apr 03 2019 *)

a(n) = fromdigits(vector(prime(n)-1, k, !isprime(k)), 10); \\ 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))))
print([a(n) for n in range(1, 14)]) # 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))
        if isprime(k+1):
            yield an
print(list(islice(agen(), 13))) # Michael S. Branicky, Jan 10 2022

A118255 a(1)=1, then a(n)=2a(n-1) if n is prime, a(n)=2a(n-1)+1 if n not prime.

Original entry on oeis.org

1, 2, 4, 9, 18, 37, 74, 149, 299, 599, 1198, 2397, 4794, 9589, 19179, 38359, 76718, 153437, 306874, 613749, 1227499, 2454999, 4909998, 9819997, 19639995, 39279991, 78559983, 157119967, 314239934, 628479869, 1256959738, 2513919477, 5027838955, 10055677911

Offset: 1

Pierre CAMI, Apr 19 2006

nonn

In base 2 a(n) is the concatenation for i=1 to n of A005171(i).

			a(2) = 2*1 = 2 as 2 is prime;
a(3) = 2*2 = 4 as 3 is prime;
a(4) = 2*4+1 = 9 as 4 is composite;
a(5) = 2*9 = 18 as 5 is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..3322 (terms 1..1000 from Harvey P. Dale)

Cf. A000225, A005171, A051006, A072762, A118256, A118257.

f:=proc(n) option remember; if n=1 then RETURN(1); fi; if isprime(n) then 2*f(n-1) else 2*f(n-1)+1; fi; end; # N. J. A. Sloane

nxt[{n_,a_}]:={n+1,If[PrimeQ[n+1],2a,2a+1]}; Transpose[NestList[nxt,{1,1},40]][[2]] (* Harvey P. Dale, Jan 22 2015 *)
Array[FromDigits[#, 2] &@ Array[Boole[! PrimeQ@ #] &, #] &, 34] (* Michael De Vlieger, Nov 01 2016 *)

from sympy import isprime, prime
def a(n): return int("".join(str(1-isprime(i)) for i in range(1, n+1)), 2)
print([a(n) for n in range(1, 35)]) # 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))
        yield an
print(list(islice(agen(), 34))) # Michael S. Branicky, Jan 10 2022

a(n) = floor(k * 2^n) where k = 0.585317... = 1 - A051006. [Charles R Greathouse IV, Dec 27 2011]
From Ridouane Oudra, Aug 26 2019: (Start)
a(n) = 2^n - 1 - (1/2)*(pi(n) + Sum_{i=1..n} 2^(n-i)*pi(i)), where pi = A000720
a(n) = A000225(n) - A072762(n). (End)

Corrected by Omar E. Pol, Nov 08 2007

Corrections verified by N. J. A. Sloane, Nov 17 2007

A139103 Numbers that show 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

10, 100, 10010, 1001010, 10010101110, 1001010111010, 10010101110101110, 1001010111010111010, 10010101110101110101110, 10010101110101110101110111110, 1001010111010111010111011111010, 1001010111010111010111011111010111110, 10010101110101110101110111110101111101110

Offset: 1

Omar E. Pol, Apr 08 2008

a(n) has A000040(n) digits, n digits "0" and A000040(n)-n digits "1".

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Binary representation of A139104.
Subset of A118256.
Cf. A139101, A139102, A139119, A139120, A139122.

Table[ sum = 0; For[i = 1, i <= Prime[n] , i++, sum = sum*2;
If[! PrimeQ[i], sum++]]; IntegerString[sum, 2], {n, 1, 20}] (* Robert Price, Apr 03 2019 *)

a(n) = fromdigits(vector(prime(n), k, !isprime(k)), 10); \\ Michel Marcus, Apr 04 2019

a(12)-a(13) from Robert Price, Apr 03 2019

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

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

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

A139101 Numbers that show the distribution of prime numbers up to the n-th prime minus 1, using "0" for primes and "1" for nonprime numbers.

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A118255 a(1)=1, then a(n)=2*a(n-1) if n is prime, a(n)=2*a(n-1)+1 if n not prime.

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A139103 Numbers that show the distribution of prime numbers up to the n-th prime using "0" for primes and "1" for nonprime numbers.

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

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

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

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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.

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A118255 a(1)=1, then a(n)=2a(n-1) if n is prime, a(n)=2a(n-1)+1 if n not prime.