A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime.

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

Offset: 1

Author

Zhi-Wei Sun, Dec 14 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>3.

This has been verified for n up to 10^8, and it is stronger than A. Murthy's conjecture related to A109909.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023

The conjecture implies the twin prime conjecture for the following reason: If x_1<...

Zhi-Wei Sun also made some similar conjectures. For example, any integer n>2 not equal to 63 can be written as x+y (x>0, y>0) with 2x-1, 2x+1 and 2xy+1 all prime.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023

Examples

			a(22)=1 since 22=4+18 with 3*4-1, 3*4+1 and 4*18-1 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Crossrefs

Cf. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[3k-1]==True&&PrimeQ[3k+1]==True&&PrimeQ[k(n-k)-1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

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