A268477 - OEIS

%I #20 Feb 11 2017 02:29:11

%S 2,3,7,31,37,41,43,53,73,83,89,101,107,109,127,137,139,151,157,163,

%T 167,179,197,211,229,233,269,281,283,307,311,313,317,353,359,367,379,

%U 389,397,401,409,419,431,433,439,443,457,461,467,491,521,523,541,547,563

%N Balanced odious primes: primes with an odd number of runs of 1's in their binary expansion.

%C Primes from A268415.

%C According to our 2007-conjecture, if pi_1(m) is the number of evil primes (A027699) not exceeding m and pi_2(m) is the number of odious primes (A027697) not exceeding m, then pi_1(m)<=pi_2(m) for all natural m except m=5 and m=6.

%C In the "balance" case of A268476,A268477, most likely, none of two types of primes

%C is in the majority beginning with any place.

%H Chai Wah Wu, <a href="/A268477/b268477.txt">Table of n, a(n) for n = 1..10000</a>

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1603.04434">Two analogs of Thue-Morse sequence</a>, arXiv:1603.04434 [math.NT], 2016.

%t Select[Prime@ Range@ 108, OddQ@ Length[Split@ IntegerDigits[#, 2] /. {0, ___} -> Nothing] &] (* _Michael De Vlieger_, Feb 08 2016 *)

%o (Python)

%o from sympy import prime

%o A268477_list = [p for p in (prime(i) for i in range(1,10**6)) if len(list(filter(bool,format(p,'b').split('0')))) % 2] # _Chai Wah Wu_, Mar 01 2016

%Y Cf. A027697, A268415, A268476.

%K nonn,base

%O 1,1

%A _Vladimir Shevelev_, Feb 05 2016

%E More terms from _Peter J. C. Moses_, Feb 05 2016