

A219052


Number of ways to write n = p + q(3  (1)^n)/2 with q <= n/2 and p, q, p^2 + q^2  1 all prime.


7



0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 0, 2, 1, 0, 0, 1, 1, 3, 0, 1, 1, 1, 1, 3, 1, 1, 4, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 4, 0, 0, 3, 0, 1, 2, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 4, 2, 1, 2, 1, 1, 0, 4, 2, 1, 1, 1, 2, 5, 4, 1, 3, 1, 1, 4, 1, 1, 2, 2
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OFFSET

1,22


COMMENTS

Conjecture: a(n) > 0 for all n > 784.
This conjecture implies Goldbach's conjecture, Lemoine's conjecture, and that there are infinitely many primes of the form p^2 + q^2  1 with p and q both prime.
It has been verified for n up to 10^8.
ZhiWei Sun also made the following general conjecture: Let d be any odd integer not congruent to 1 modulo 3. Then, all large even numbers can be written as p + q with p, q, p^2 + q^2 + d all prime. If d is also not divisible by 5, then all large odd numbers can be represented as p + 2q with p, q, p^2 + q^2 + d all prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..20000
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.


EXAMPLE

a(12) = 1 since {5, 7} is the only prime pair {p, q} for which p + q = 12, and p^2 + q^2  1 is prime.


MATHEMATICA

a[n_] := a[n] = Sum[If[PrimeQ[n  (1 + Mod[n, 2])Prime[k]] == True && PrimeQ[Prime[k]^2 + (n  (1 + Mod[n, 2])Prime[k])^2  1] == True, 1, 0], {k, 1, PrimePi[n/2]}]; Do[Print[n, " ", a[n]], {n, 1, 20000}]


CROSSREFS

Cf. A000040, A002375, A046927, A218754, A218585, A218654, A218825, A219023, A219026.
Sequence in context: A065252 A115211 A097516 * A060826 A078134 A282380
Adjacent sequences: A219049 A219050 A219051 * A219053 A219054 A219055


KEYWORD

nonn,nice


AUTHOR

ZhiWei Sun, Nov 10 2012


STATUS

approved



