%I #30 Jun 27 2022 21:18:26
%S 3,5,17,23,29,43,53,71,83,89,101,113,139,149,163,197,257,263,269,277,
%T 281,293,311,317,337,347,349,353,359,373,383,389,401,449,461,467,479,
%U 503,509,523,547,571,593,599,619,643,673,683,691,739,751,773,797,811
%N Evil primes: primes with even number of 1's in their binary expansion.
%C Comment from _Vladimir Shevelev_, Jun 01 2007: Conjecture: If pi_1(m) is the number of a(n) not exceeding m and pi_2(m) is the number of A027697(n) not exceeding m then pi_1(m) <= smaller than pi_2(m) for all natural m except m=5 and m=6. I verified this conjecture up to 10^9. Moreover I conjecture that pi_2(m)-pi_1(m) tends to infinity with records at the primes m=2, 13, 41, 61, 67, 79, 109, 131, 137, ...
%H T. D. Noe, <a href="/A027699/b027699.txt">Table of n, a(n) for n = 1..10000</a>
%H E. Fouvry, C. Mauduit, <a href="http://dx.doi.org/10.1007/BF01444238">Sommes des chiffres et nombres presque premiers</a>, (French) [Sums of digits and almost primes] Math. Ann. 305 (1996), no. 3, 571--599. MR1397437 (97k:11029).
%H V. 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.
%t Select[Prime[Range[200]], EvenQ[Count[IntegerDigits[ #,2],1]]&] (* _T. D. Noe_, Jun 12 2007 *)
%o (PARI) forprime(p=1,999,norml2(binary(p))%2 || print1(p","))
%o (PARI) isA027699(p)=isprime(p) && !bittest(norml2(binary(p)),0) \\ _M. F. Hasler_, Dec 12 2010
%o (Python)
%o from sympy import isprime
%o def ok(n): return bin(n).count("1")%2 == 0 and isprime(n)
%o print([k for k in range(812) if ok(k)]) # _Michael S. Branicky_, Jun 27 2022
%Y Cf. A027697, A066148, A066149.
%Y Cf. A001969 (evil numbers), A129771 (evil odd numbers)
%Y Cf. A130911 (prime race between evil primes and odious primes).
%K nonn,easy,base
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Erich Friedman_