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 A130911 a(n) = number of primes with odd binary weight among the first n primes minus the number with an even binary weight. 5
 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, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 21, 20, 19, 20, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Prime race between evil primes (A027699) and odious primes (A027697). 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 a(n) = (number of odious primes <= prime(n)) - (number of evil primes <= prime(n)). a(n) = A200247(n) - A200246(n). 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 CROSSREFS Cf. A095005, A095006. Cf. A199399, A027697, A027698, A027699, A027700, A200244, A200245, A200246, A200247. Cf. A156549 (race between primes having an odd/even number of zeros in binary). Sequence in context: A035224 A272677 A165579 * A319300 A118829 A118832 Adjacent sequences:  A130908 A130909 A130910 * A130912 A130913 A130914 KEYWORD nice,sign,base AUTHOR T. D. Noe, Jun 08 2007 EXTENSIONS Edited by N. J. A. Sloane, Nov 16 2011 STATUS approved

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Last modified June 17 17:12 EDT 2019. Contains 324196 sequences. (Running on oeis4.)