

A230219


Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.


14



0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 5, 2, 2, 4, 3, 1, 4, 3, 1, 4, 1, 2, 5, 2, 3, 4, 3, 3, 8, 6, 3, 12, 6, 2, 13, 3, 3, 7, 6, 4, 5, 4, 4, 8, 7, 4, 12, 7, 3, 19, 6, 3, 16, 5, 4, 9, 5, 5, 7, 10, 4, 5, 8, 3, 14, 4, 3, 14, 2, 5, 12, 5, 2, 14, 9, 2, 10, 12, 4, 12, 7, 6, 12, 7, 9, 14, 8, 6, 12, 5, 4, 19, 8, 4, 23, 6, 3, 14
(list;
graph;
refs;
listen;
history;
text;
internal format)



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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei 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+1Prime[i]Prime[j]], 1, 0], {i, 1, PrimePi[n1]}, {j, i, PrimePi[2n2Prime[i]]}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A002375, A068307, A230217, A230140, A230141.
Sequence in context: A165195 A121487 A057031 * A147292 A078391 A153206
Adjacent sequences: A230216 A230217 A230218 * A230220 A230221 A230222


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 11 2013


STATUS

approved



