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

1,1

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A027697, A066148, A066149.

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

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

Sequence in context: A218624 A152078 A152079 * A153417 A069687 A079017

Adjacent sequences: A027696 A027697 A027698 * A027700 A027701 A027702

Keyword

nonn,easy,base

Author

N. J. A. Sloane

Extensions

More terms from Erich Friedman

Status

approved