

A227908


Number of ways to write 2*n = p + q with p, q and (p1)^2 + q^2 all prime.


6



0, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 3, 2, 2, 0, 2, 6, 1, 3, 5, 2, 3, 2, 1, 2, 2, 5, 4, 3, 2, 3, 8, 1, 4, 3, 3, 2, 5, 1, 2, 4, 5, 3, 4, 4, 2, 6, 1, 4, 5, 3, 3, 6, 2, 6, 5, 4, 5, 7, 3, 1, 9, 2, 3, 6, 1, 2, 5, 4, 7, 2, 7, 6, 6, 2, 4, 10, 3, 3, 6, 1, 7, 9, 5, 4, 5, 4, 3, 5, 3, 5, 8, 4, 4, 5, 2, 11, 9, 4
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OFFSET

1,5


COMMENTS

Conjecture: a(n) > 0 except for n = 1, 16, 292.
This implies not only Goldbach's conjecture for even numbers, but also MingZhi Zhang's conjecture (cf. A036468) that any odd number greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.
We have verified the conjecture for n up to 10^7.


LINKS



EXAMPLE

a(7) = 1 since 2*7 = 11 + 3, and (111)^2 + 3^2 = 109 is prime.
a(19) = 1 since 2*19 = 7 + 31, and (71)^2 + 31^2 = 997 is prime.


MATHEMATICA

a[n_]:=Sum[If[PrimeQ[2nPrime[i]]&&PrimeQ[(Prime[i]1)^2+(2nPrime[i])^2], 1, 0], {i, 1, PrimePi[2n2]}]
Table[a[n], {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



