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A227908
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Number of ways to write 2*n = p + q with p, q and (p-1)^2 + q^2 all prime.
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6
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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
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1,5
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COMMENTS
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Conjecture: a(n) > 0 except for n = 1, 16, 292.
This implies not only Goldbach's conjecture for even numbers, but also Ming-Zhi 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.
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LINKS
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EXAMPLE
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a(7) = 1 since 2*7 = 11 + 3, and (11-1)^2 + 3^2 = 109 is prime.
a(19) = 1 since 2*19 = 7 + 31, and (7-1)^2 + 31^2 = 997 is prime.
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MATHEMATICA
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a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)^2+(2n-Prime[i])^2], 1, 0], {i, 1, PrimePi[2n-2]}]
Table[a[n], {n, 1, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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