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%I #19 Mar 24 2023 08:04:56
%S 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 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 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 Race between primes having an odd/even number of zeros in their binary representation.
%C 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 T. D. Noe, <a href="/A156549/b156549.txt">Table of n, a(n) for n = 1..10000</a>
%F 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 cnt=0; Table[p=Prime[n]; If[OddQ[Count[IntegerDigits[p,2],0]], cnt++, cnt-- ]; cnt, {n,100}]
%o (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 Cf. A066148, A066149, A130911.
%K nonn,base
%O 1,6
%A _T. D. Noe_, Feb 09 2009
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