A027699 Evil primes: primes with even number of 1's in their binary expansion.

3, 5, 17, 23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 257, 263, 269, 277, 281, 293, 311, 317, 337, 347, 349, 353, 359, 373, 383, 389, 401, 449, 461, 467, 479, 503, 509, 523, 547, 571, 593, 599, 619, 643, 673, 683, 691, 739, 751, 773, 797, 811

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• base

Comments

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, ...

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• E. Fouvry, C. Mauduit, Sommes des chiffres et nombres presque premiers, (French) [Sums of digits and almost primes] Math. Ann. 305 (1996), no. 3, 571--599. MR1397437 (97k:11029).
• V. Shevelev, A conjecture on primes and a step towards justification, arXiv:0706.0786 [math.NT], 2007.

Crossrefs

Cf. A027697, A066148, A066149.

Cf. A001969 (evil numbers), A129771 (evil odd numbers)

Cf. A130911 (prime race between evil primes and odious primes).

Programs

Mathematica

Select[Prime[Range[200]], EvenQ[Count[IntegerDigits[ #,2],1]]&] (* T. D. Noe, Jun 12 2007 *)

PARI

forprime(p=1,999,norml2(binary(p))%2 || print1(p","))

PARI

isA027699(p)=isprime(p) && !bittest(norml2(binary(p)),0) \\ M. F. Hasler, Dec 12 2010

Python

from sympy import isprime
def ok(n): return bin(n).count("1")%2 == 0 and isprime(n)
print([k for k in range(812) if ok(k)]) # Michael S. Branicky, Jun 27 2022

Extensions

More terms from Erich Friedman