A095018 a(n) is the number of primes p which have exactly n zeros and n ones when written in binary.

1, 0, 2, 4, 17, 28, 189, 531, 1990, 5747, 23902, 76658, 291478, 982793, 3677580, 13214719, 49161612, 177190667, 664806798, 2443387945

Offset: 1

Author

Antti Karttunen, Jun 01 2004

Keywords

• nonn
• base

Comments

a(n) is the number of terms in A066196 which lie between 2^(2n-1) and 2^2n inclusively.

Examples

			a(1) = 1 since only 2_10 = 10_2 satisfies the criterion;
a(2) = 0 since there is no prime between 4 and 16 which meets the criterion.
The only primes in the range ]2^5,2^6[ with equal numbers of ones and zeros in their binary expansion are 37 (in binary 100101) and 41 (in binary 101011) thus a(3)=2.
a(4) = 4 since 139, 149, 163 and 197 meet the criterion; etc.
		

Links

• Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
• Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Crossrefs

Cf. A066196, A095005, A095006, A095052, A095053, A280872.

Programs

Mathematica

f[n_] := Block[{c = 0, p = NextPrime[2^(2n -1) -1], lmt = 2^(2n)}, While[p < lmt, If[DigitCount[p, 2, 1] == n, c++]; p = NextPrime@ p]; c]; Array[f, 17] (* K. D. Bajpai and Robert G. Wilson v, Jan 10 2017 *)

Python

from itertools import combinations
from sympy import isprime
def A095018(n): return sum(1 for d in combinations((1<Chai Wah Wu, Jul 18 2025

Extensions

Edited by N. J. A. Sloane, Jan 16 2017

a(18)-a(20) from Amiram Eldar, Nov 21 2020