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

1,5

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<...<x_k are positive integers and q_1,...,q_k are distinct primes greater than x_k, then by the Chinese Remainder Theorem there are infinitely many positive integers n such that x_i(n-x_i) == 1 (mod q_i).

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

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.

Example

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

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}]

Crossrefs

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

Sequence in context: A305976 A074592 A089993 * A351284 A268317 A234092

Adjacent sequences: A220428 A220429 A220430 * A220432 A220433 A220434

Keyword

nonn

Author

Zhi-Wei Sun, Dec 14 2012

Status

approved