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

Author

Zhi-Wei Sun, Oct 13 2013

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

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

Programs

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}]