A156549 Race between primes having an odd/even number of zeros in their binary representation.

1, 0, 1, 0, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 7, 8, 9, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 21, 22, 21, 22, 21, 22, 21, 22, 23, 24, 25, 26, 25, 26, 25, 26, 27, 26, 27, 26, 25, 24, 23, 22

Offset

1,6

Comments

See A066148 and A066149 for primes with an even/odd number of zeros in their binary representation. Sequence A130911 shows the race between primes having an odd/even number of ones in their binary representation. In this sequence (and A130911), it appears that the primes with an odd number of zeros (or ones) dominate the primes with an even number of zeros (or ones). In general, it appears that the sequences grow for primes having an even number of bits and "rest" for primes having an odd number of bits. [Corrected by Benjamin Chaffin, Jun 11 2026]

Odd-zeros continue to lead through 10^19. The growth is identical to A130911 for primes with an even number of bits (where the parity of 1s and 0s is the same, and the sequence grows), and inverse to A130911 for primes with an odd number of bits (where the parity is opposite, and the sequence "rests"). - Benjamin Chaffin, Jun 11 2026

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Benjamin Chaffin, Graph of odious vs. evil and odd-zero vs. even-zero primes from 0..2^63

Formula

a(n) = (number of primes having an odd number of zeros <= prime(n)) - (number of primes having an even number of zeros <= prime(n)).

Mathematica

cnt=0; Table[p=Prime[n]; If[OddQ[Count[IntegerDigits[p, 2], 0]], cnt++, cnt-- ]; cnt, {n, 100}]
(* Alternative: *)
Accumulate[Table[If[OddQ[DigitCount[p, 2, 0]], 1, -1], {p, Prime[Range[90]]}]] (* Harvey P. Dale, Apr 02 2025 *)

Prog

(PARI) f(p)={v=binary(p); s=0; for(k=1, #v, if(v[k]==0, s++)); return(s%2)}; nO=0; nE=0; forprime(p=2, 435, if(f(p), nO++, nE++); an = nO-nE; print1(an, ", ")) \\ Washington Bomfim, Jan 14 2011

Crossrefs

Cf. A066148, A066149, A130911.

Sequence in context: A173523 A199323 A383037 * A275868 A100795 A045781

Adjacent sequences: A156546 A156547 A156548 * A156550 A156551 A156552

Keyword

nonn,base

Author

T. D. Noe, Feb 09 2009

Status

approved