OFFSET
1,10
COMMENTS
Conjecture: a(n) > 0 for all n > 6.
This implies Goldbach's weak conjecture for odd numbers and also Goldbach's conjecture for even numbers.
The conjecture also implies that there are infinitely many primes in A230217.
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.
EXAMPLE
a(18) = 1 since 2*18 + 1 = 7 + 13 + 17, and 7, 13, 17 are terms of A230217, and 7 + 13 + 9 = 29 is a prime.
MATHEMATICA
RQ[n_]:=PrimeQ[n+6]&&PrimeQ[3n+8]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&PrimeQ[Prime[i]+Prime[j]+9]&&SQ[2n+1-Prime[i]-Prime[j]], 1, 0], {i, 1, PrimePi[n-1]}, {j, i, PrimePi[2n-2-Prime[i]]}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 11 2013
STATUS
approved