A130911 - OEIS

%I #29 Dec 22 2021 02:18:24

%S 1,0,-1,0,1,2,1,2,1,0,1,2,3,2,3,2,3,4,5,4,5,6,5,4,5,4,5,6,7,6,7,8,9,8,

%T 7,8,9,8,9,10,11,12,13,14,13,14,15,16,17,18,19,20,21,22,21,20,19,20,

%U 19,18,19,18,19,18,19,18,19,18,17,16,15,14,15,14,15,14,13,14,13,14,15,16,17,18,19,20,19,20,19,20,19,18,19,20,21,20,19

%N a(n) is the number of primes with odd binary weight among the first n primes minus the number with an even binary weight.

%C Prime race between evil primes (A027699) and odious primes (A027697).

%C Shevelev conjectures that a(n) >= 0 for n > 3. Surprisingly, the conjecture also appears to be true if we count zeros instead of ones in the binary representation of prime numbers.

%C The conjecture is true for primes up to at least 10^13. Mauduit and Rivat prove that half of all primes are evil. - _T. D. Noe_, Feb 09 2009

%H T. D. Noe, <a href="/A130911/b130911.txt">Table of n, a(n) for n = 1..10000</a>

%H CNRS Press release, <a href="http://www2.cnrs.fr/en/1732.htm">The sum of digits of prime numbers is evenly distributed</a>, May 12, 2010.

%H Christian Mauduit and Joël Rivat, <a href="http://annals.math.princeton.edu/2010/171-3/p04">Sur un problème de Gelfond: la somme des chiffres des nombres premiers</a>, Annals Math., 171 (2010), 1591-1646.

%H ScienceDaily, <a href="http://www.sciencedaily.com/releases/2010/05/100512172533.htm">Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis</a>, May 12, 2010.

%H Vladimir Shevelev, <a href="https://arxiv.org/abs/0706.0786">A conjecture on primes and a step towards justification</a>, arXiv:0706.0786 [math.NT], 2007.

%H Vladimir Shevelev, <a href="https://www.arxiv.org/abs/0707.1761">On excess of odious primes</a>, arXiv:0707.1761 [math.NT], 2007.

%F a(n) = (number of odious primes <= prime(n)) - (number of evil primes <= prime(n)).

%F a(n) = A200247(n) - A200246(n).

%t cnt=0; Table[p=Prime[n]; If[EvenQ[Count[IntegerDigits[p,2],1]], cnt--, cnt++ ]; cnt, {n,10000}]

%t Accumulate[If[OddQ[DigitCount[#,2,1]],1,-1]&/@Prime[Range[100]]] (* _Harvey P. Dale_, Aug 09 2013 *)

%o (PARI)f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==1,s++)); return(s%2)};nO=0;nE=0;forprime(p=2,520,if(f(p),nO++, nE++);an=nO-nE;print1(an,", ")) \\ _Washington Bomfim_, Jan 14 2011

%o (Python)

%o from sympy import nextprime

%o from itertools import islice

%o def agen():

%o     p, evod = 2, [0, 1]

%o     while True:

%o         yield evod[1] - evod[0]

%o         p = nextprime(p); evod[bin(p).count('1')%2] += 1

%o print(list(islice(agen(), 97))) # _Michael S. Branicky_, Dec 21 2021

%Y Cf. A095005, A095006.

%Y Cf. A199399, A027697, A027698, A027699, A027700, A200244, A200245, A200246, A200247.

%Y Cf. A156549 (race between primes having an odd/even number of zeros in binary).

%K nice,sign,base

%O 1,6

%A _T. D. Noe_, Jun 08 2007

%E Edited by _N. J. A. Sloane_, Nov 16 2011