OFFSET
1,6
COMMENTS
Conjecture: a(n) > 0 for all n > 1.
This has been verified for n up to 10^8. It implies that there are infinitely many primes of the form x^2 + x + 1.
The author also guesses that any integer n > 1157 can be written as x + y with x and y positive integers, and (x*y)^2 + x*y + 1 and (x*y)^2 + x*y - 1 twin primes.
Zhi-Wei Sun has made the following general conjecture: For each prime p, any sufficiently large integer n can be written as x + y, where x and y are positive integers with ((x*y)^p - 1)/(x*y - 1) prime. (For p = 5, 7 it suffices to require n > 28 and n > 46 respectively.)
Compare this with Sun's another conjecture related to A219791.
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(49) = 1 since 49 = 3 + 46 with (3*46)^2 + 3*46 + 1 = 19183 prime.
MATHEMATICA
a[n_] := a[n] = Sum[If[PrimeQ[k(n - k)(k(n - k) + 1) + 1] == True, 1, 0], {k, 1, n/2}]; Do[Print[n, " ", a[n]], {n, 1, 10000}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 29 2012
STATUS
approved