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

0, 1, 2, 2, 3, 4, 5, 4, 5, 4, 7, 7, 7, 8, 10, 8, 10, 9, 11, 13, 14, 12, 17, 14, 14, 13, 16, 16, 20, 19, 20, 21, 22, 17, 22, 23, 22, 26, 28, 25, 31, 24, 27, 31, 32, 28, 37, 32, 33, 34, 33, 35, 39, 38, 41, 44, 46, 35, 45, 43, 43, 46, 48, 45, 55, 45, 46, 56, 53, 51

1

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

S. Chowla, The representation of a number as four squares and a prime, Acta Arithmetica, Vol. 1, No. 1, pp. 115-122.

(Mma) nn = 10; s = Select[Sort[Flatten[Table[x^2 + y^2 + z^2 + u^2 + p, {x, 0, nn}, {y, x, nn}, {z, y, nn}, {u, z, 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-S000136 (2 to 4 squares), A002471, A064272 (1 square).

nonn

T. D. Noe, Jul 09 2014