A277688 Odd numbers k such that there is no prime p < k/2 with k - 2*p and k + 2*p both prime.

1, 3, 5, 19, 29, 31, 43, 49, 55, 59, 61, 71, 79, 83, 89, 91, 101, 109, 113, 115, 119, 125, 127, 131, 139, 149, 151, 155, 161, 163, 167, 169, 175, 179, 191, 193, 197, 199, 203, 209, 211, 215, 223, 227, 229, 239, 241, 247, 251, 253, 259, 265, 269, 271, 281, 283

Offset: 1

Author

Vladimir Shevelev, Apr 11 2017

Keywords

• nonn

Comments

Or, odd integers k such that k + 2*p is composite for all primes p, q with 2*p + q = k. By the Lemoine-Levy conjecture, for every odd k>5, there are primes p and q such that k=2*p+q. Numbers 1,3,5 formally satisfy the condition.

The sequence is an analog of A284919 for odd numbers.

Conjecture: k=59 and k=151 are the only terms k>5 satisfying the additional condition that k + 2*q is composite for every prime p,q such that 2*p+q=k.

This conjecture arose from the calculations up to 500001 by Peter J. C. Moses and confirmed by M. F. Hasler.

More than half of all odd numbers are in this sequence: for k < 2000, the percentage is below 50%, but for k < 1e4, 2e4 and 4e4 the percentage is > 55%, 56% and 58%, respectively. - M. F. Hasler, Apr 11 2017

Crossrefs

Cf. A046927, A284919.

Programs

Mathematica

Select[Range[1, 283, 2], Total@ Boole@ Map[Function[p, Times @@ Boole@ Map[PrimeQ, {# - 2 p, # + 2 p}] == 1], Prime@ Range@ PrimePi[#/2]] == 0 &] (* Michael De Vlieger, Apr 22 2017 *)

PARI

is(k)=bittest(k,0)&&!forprime(p=2,k\2,(isprime(k-2*p)&&isprime(k+2*p))&&return) \\ M. F. Hasler, Apr 11 2017

Extensions

More terms from Peter J. C. Moses, Apr 11 2017