A268477 Balanced odious primes: primes with an odd number of runs of 1's in their binary expansion.

2, 3, 7, 31, 37, 41, 43, 53, 73, 83, 89, 101, 107, 109, 127, 137, 139, 151, 157, 163, 167, 179, 197, 211, 229, 233, 269, 281, 283, 307, 311, 313, 317, 353, 359, 367, 379, 389, 397, 401, 409, 419, 431, 433, 439, 443, 457, 461, 467, 491, 521, 523, 541, 547, 563

Offset: 1

Author

Vladimir Shevelev, Feb 05 2016

Keywords

• nonn
• base

Comments

Primes from A268415.

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.

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

is in the majority beginning with any place.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Crossrefs

Cf. A027697, A268415, A268476.

Programs

Mathematica

Select[Prime@ Range@ 108, OddQ@ Length[Split@ IntegerDigits[#, 2] /. {0, _} -> Nothing] &] (* Michael De Vlieger, Feb 08 2016 *)

Python

from sympy import prime
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

Extensions

More terms from Peter J. C. Moses, Feb 05 2016