A227909 Number of ways to write 2*n = p + q with p, q and (p-1)*(q+1) - 1 all prime.

0, 1, 1, 1, 2, 1, 2, 3, 3, 1, 2, 5, 2, 3, 2, 3, 3, 5, 3, 1, 5, 4, 5, 4, 3, 4, 7, 4, 4, 2, 1, 4, 9, 2, 4, 11, 4, 2, 6, 2, 6, 11, 6, 4, 3, 3, 5, 6, 4, 3, 6, 2, 4, 10, 3, 10, 12, 7, 1, 6, 6, 5, 11, 4, 5, 6, 4, 3, 11, 2, 10, 13, 4, 6, 5, 2, 14, 13, 2, 2, 5, 5, 9, 15, 5, 3, 7, 8, 5, 3, 5, 7, 15, 3, 1, 8, 5, 7, 11, 4

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 1.

This is stronger than Goldbach's conjecture for even numbers. It also implies A. Murthy's conjecture (cf. A109909) for even numbers.

We have verified the conjecture for n up to 2*10^7.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 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(6) = 1 since 2*6 = 5 + 7, and (5-1)*(7+1)-1 = 31 is prime.
a(10) = 1 since 2*10 = 7 + 13, and (7-1)*(13+1)-1 = 83 is prime.
a(20) = 1 since 2*20 = 17 + 23, and (17-1)*(23+1)-1 = 383 is prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)(2n-Prime[i]+1)-1], 1, 0], {i, 1, PrimePi[2n-2]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A002375, A109909, A218754, A219052, A220431, A220455, A220554, A227908, A230230.

Sequence in context: A376238 A348330 A328471 * A301984 A210805 A303842

Adjacent sequences: A227906 A227907 A227908 * A227910 A227911 A227912

Keyword

nonn

Author

Olivier Gérard and Zhi-Wei Sun, Oct 13 2013

Status

approved