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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
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.
Zhi-Wei 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
Zhi-Wei 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}]
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Nov 10 2012
STATUS
approved