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

Author

T. D. Noe, Feb 09 2009

Keywords

• nonn
• base

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 odd number of bits and "rest" for primes having an even number of bits.

Links

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

Crossrefs

Cf. A066148, A066149, A130911.

Programs

Mathematica

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

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

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))