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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 14 2012
STATUS
approved