A230502 Number of ways to write n = (2-(n mod 2))*p + q + r with p <= q <= r such that p, q, r, p^2 - 2, q^2 - 2, r^2 - 2 are all prime.

0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 4, 3, 4, 2, 2, 3, 2, 2, 4, 2, 3, 4, 4, 3, 3, 3, 3, 4, 5, 4, 4, 3, 3, 5, 7, 5, 6, 5, 5, 5, 6, 3, 5, 5, 5, 5, 6, 4, 4, 4, 5, 6, 7, 5, 6, 4, 3, 5, 7, 5, 5, 7, 7, 6, 7, 4, 6, 6, 7, 7, 6, 4, 6, 4, 4, 8, 8, 6, 6, 7, 6, 6, 10

Offset

1,9

Comments

Conjecture: a(n) > 0 for all n > 6.

This is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013. It also implies that there are infinitely many primes p with p^2 - 2 also prime.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023

Links

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

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(10) = 1 since 10 = 2*2 + 3 + 3 with 2, 3, 2^2 - 2 = 2, 3^2 - 2 = 7 all prime.
a(19) = 2 since 19 = 3 + 3 + 13 = 5 + 7 + 7 with 3, 13, 5, 7, 3^2 - 2 = 7, 13^2 - 2 = 167, 5^2 - 2 = 23, 7^2 - 2 = 47 all prime.

Mathematica

pp[n_]:=PrimeQ[n^2-2]
pq[n_]:=PrimeQ[n]&&pp[n]
a[n_]:=Sum[If[pp[Prime[i]]&&pp[Prime[j]]&&pq[n-(2-Mod[n, 2])Prime[i]-Prime[j]], 1, 0], {i, 1, PrimePi[n/(4-Mod[n, 2])]}, {j, i, PrimePi[(n-(2-Mod[n, 2])Prime[i])/2]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A062326, A068307, A230219, A230351, A230493, A230494.

Sequence in context: A340134 A218469 A343333 * A280253 A008677 A036497

Adjacent sequences: A230499 A230500 A230501 * A230503 A230504 A230505

Keyword

nonn

Author

Zhi-Wei Sun, Oct 21 2013

Status

approved