A219864 Number of ways to write n as p+q with p and 2pq+1 both prime

0, 0, 1, 1, 2, 3, 0, 2, 4, 2, 2, 4, 1, 2, 6, 3, 1, 2, 2, 5, 3, 1, 1, 7, 2, 6, 3, 1, 6, 8, 2, 2, 5, 3, 3, 8, 2, 4, 6, 3, 4, 4, 1, 3, 7, 2, 3, 7, 3, 6, 8, 2, 1, 12, 5, 4, 7, 4, 7, 7, 7, 5, 4, 4, 6, 9, 2, 2, 13, 2, 5, 7, 2, 4, 18, 6, 3, 5, 6, 5, 8, 4, 2, 9, 4, 10, 5, 2, 5, 17, 3, 3, 7, 7, 5, 8, 3, 3, 17, 8

Offset

1,5

Comments

Conjecture: a(n)>0 for all n>7.

This has been verified for n up to 3*10^8.

Zhi-Wei Sun also made the following general conjecture: For each odd integer m not congruent to 5 modulo 6, any sufficiently large integer n can be written as p+q with p and 2*p*q+m both prime.

For example, when m = 3, -3, 7, 9, -9, -11, 13, 15, it suffices to require that n is greater than 1, 29, 16, 224, 29, 5, 10, 52 respectively.

Sun also guessed that any integer n>4190 can be written as p+q with p, 2*p*q+1, 2*p*q+7 all prime, and any even number n>1558 can be written as p+q with p, q, 2*p*q+3 all prime. He has some other similar observations.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(10)=2 since 10=3+7=7+3 with 2*3*7+1=43 prime.
a(263)=1 since 83 is the only prime p with 2p(263-p)+1 prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k](n-Prime[k])+1]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 1000}]

Crossrefs

Cf. A219842, A002372, A046927, A219838, A219791, A219782, A036468.

Sequence in context: A369287 A267852 A328568 * A257844 A194745 A248342

Adjacent sequences: A219861 A219862 A219863 * A219865 A219866 A219867

Keyword

nonn

Author

Zhi-Wei Sun, Nov 30 2012

Status

approved