Number of ways that n can be written as the sum of two nonnegative squares and a prime.

0, 1, 3, 3, 2, 4, 6, 4, 3, 3, 6, 8, 5, 4, 8, 6, 5, 8, 7, 9, 9, 8, 10, 8, 5, 4, 13, 12, 8, 10, 10, 12, 11, 7, 6, 14, 10, 10, 17, 8, 10, 16, 13, 12, 13, 10, 16, 18, 10, 8, 14, 15, 14, 18, 15, 16, 19, 12, 10, 18, 12, 12, 25, 12, 11, 20, 17, 20, 20, 16, 18, 24, 14

1

T. D. Noe, Plot of 10000 terms

T. D. Noe, Table of 10000 terms

(Mma) nn = 10; s = Select[Sort[Flatten[Table[x^2 + y^2 + p, {x, 0, nn}, {y, 0, nn}, {p, Prime[Range[PrimePi[nn^2]]]}]]], # <= nn^2 &]; t = Table[0, {nn^2}]; s = Tally[s]; Do[t[[i[[1]]]] = i[[2]], {i, s}]; t

Cf. S000125-S000135 (2 to 4 squares), A002471, A064272 (1 square).

nonn

T. D. Noe, Jul 11 2014