This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A156549 #21 Apr 02 2025 18:40:21
%S A156549 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,
%T A156549 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,
%U A156549 21,22,21,22,21,22,21,22,23,24,25,26,25,26,25,26,27,26,27,26,25,24,23,22
%N A156549 Race between primes having an odd/even number of zeros in their binary representation.
%C A156549 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 odd number of bits and "rest" for primes having an even number of bits.
%H A156549 T. D. Noe, <a href="/A156549/b156549.txt">Table of n, a(n) for n = 1..10000</a>
%F A156549 a(n) = (number of primes having an odd number of zeros <= prime(n)) - (number of primes having an even number of zeros <= prime(n))
%t A156549 cnt=0; Table[p=Prime[n]; If[OddQ[Count[IntegerDigits[p,2],0]], cnt++, cnt-- ]; cnt, {n,100}]
%t A156549 Accumulate[Table[If[OddQ[DigitCount[p,2,0]],1,-1],{p,Prime[Range[90]]}]] (* _Harvey P. Dale_, Apr 02 2025 *)
%o A156549 (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
%Y A156549 Cf. A066148, A066149, A130911.
%K A156549 nonn,base
%O A156549 1,6
%A A156549 _T. D. Noe_, Feb 09 2009