A066196 Primes which have an equal number of zeros and ones in their binary expansion.

2, 37, 41, 139, 149, 163, 197, 541, 557, 563, 569, 587, 601, 613, 617, 647, 653, 659, 661, 677, 709, 787, 809, 929, 2141, 2203, 2221, 2251, 2281, 2333, 2347, 2357, 2381, 2389, 2393, 2417, 2467, 2473, 2617, 2659, 2699, 2707, 2713, 2729, 2837, 2851, 2857

Offset: 1

Author

Robert G. Wilson v, Dec 15 2001

Keywords

• nonn
• base

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A000040, A031443.

Programs

Mathematica

Prime[ Select[ Range[ 10^3 ], Count[ IntegerDigits[ Prime[ # ], 2 ], 0 ] == Count[ IntegerDigits[ Prime[ # ], 2 ], 1 ] & ] ]
digBalQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ@(m = Length@d) && Count[d, 1] == m/2]; Select[Range[3000], PrimeQ[#] && digBalQ[#] &] (* Amiram Eldar, Nov 21 2020 *)
Select[Prime[Range[500]],DigitCount[#,2,1]==DigitCount[#,2,0]&] (* Harvey P. Dale, Jun 24 2025 *)

PARI

isok(p) = isprime(p) && (2*hammingweight(p) == #binary(p)); \\ Michel Marcus, May 16 2022

Python

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def agen():
    yield from filter(isprime, (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1))))
print(list(islice(agen(), 50))) # Michael S. Branicky, May 15 2022

Formula

A000040 INTERSECT A031443. - R. J. Mathar, Jun 01 2011