A230241 Number of ways to write n = p + q with p, 3*p - 10 and (p-1)*q - 1 all prime, where q is a positive integer.

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

Offset

1,8

Comments

Conjecture: a(n) > 0 for all n > 5.

This implies A. Murthy's conjecture mentioned in A109909.

We have verified the conjecture for n up to 10^8.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 29 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(9) = 1 since 9 = 7 + 2 with 7, 3*7-10 = 11, (7-1)*2-1 = 11 all prime.
a(27) = 1 since 27 = 13 + 14, and the three numbers 13, 3*13-10 = 29, (13-1)*14-1 = 167 are prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[3Prime[i]-10]&&PrimeQ[(Prime[i]-1)(n-Prime[i])-1], 1, 0], {i, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A109909, A227908, A227909, A230227, A230230.

Sequence in context: A242425 A263104 A282518 * A029315 A070080 A230196

Adjacent sequences: A230238 A230239 A230240 * A230242 A230243 A230244

Keyword

nonn

Author

Zhi-Wei Sun, Oct 13 2013

Status

approved