OFFSET
1,6
COMMENTS
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.
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
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
CNRS Press release, The sum of digits of prime numbers is evenly distributed, May 12, 2010.
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 12, 2010.
Vladimir Shevelev, A conjecture on primes and a step towards justification, arXiv:0706.0786 [math.NT], 2007.
Vladimir Shevelev, On excess of odious primes, arXiv:0707.1761 [math.NT], 2007.
FORMULA
MATHEMATICA
cnt=0; Table[p=Prime[n]; If[EvenQ[Count[IntegerDigits[p, 2], 1]], cnt--, cnt++ ]; cnt, {n, 10000}]
Accumulate[If[OddQ[DigitCount[#, 2, 1]], 1, -1]&/@Prime[Range[100]]] (* Harvey P. Dale, Aug 09 2013 *)
PROG
(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
(Python)
from sympy import nextprime
from itertools import islice
def agen():
p, evod = 2, [0, 1]
while True:
yield evod[1] - evod[0]
p = nextprime(p); evod[bin(p).count('1')%2] += 1
print(list(islice(agen(), 97))) # Michael S. Branicky, Dec 21 2021
CROSSREFS
KEYWORD
nice,sign,base
AUTHOR
T. D. Noe, Jun 08 2007
EXTENSIONS
Edited by N. J. A. Sloane, Nov 16 2011
STATUS
approved