A258141 Number of ways to write n as p^2 + q with p, q and 2*p + 3 all prime.

0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0

Offset

1,27

Comments

Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5.

We have verified this for n up to 10^8. See also A258139 for a weaker version of this conjecture.

The conjecture is somewhat similar to Goldbach's Conjecture. It implies that there are infinitely many primes p with 2*p + 3 prime.

Links

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

Example

a(11) = 1 since 11 = 2^2 + 7 with 2, 7 and 2*2 + 3 all prime.

Mathematica

Do[r=0; Do[If[PrimeQ[2Prime[k]+3]&&PrimeQ[n-Prime[k]^2], r=r+1], {k, 1, PrimePi[Sqrt[n]]}]; Print[n, " ", r]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000040, A002375, A023204, A258139, A258140.

Sequence in context: A368457 A073368 A321756 * A037889 A339872 A280269

Adjacent sequences: A258138 A258139 A258140 * A258142 A258143 A258144

Keyword

nonn

Author

Zhi-Wei Sun, May 22 2015

Status

approved