

A294081


Number of partitions of n into three squares and two nonnegative 7th powers.


0



1, 2, 3, 3, 3, 3, 3, 2, 2, 3, 4, 4, 3, 3, 3, 2, 2, 3, 5, 5, 4, 3, 3, 2, 2, 3, 5, 6, 4, 4, 3, 3, 2, 3, 5, 5, 5, 4, 5, 3, 3, 4, 5, 5, 3, 4, 4, 3, 2, 3, 6, 7, 6, 5, 6, 5, 4, 3, 4, 5, 3, 4, 4, 4, 3, 4, 7, 7, 6, 5, 5, 3, 3, 4, 7, 7, 6, 5, 4, 3, 2, 5, 7, 8, 5, 5, 6
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OFFSET

0,2


COMMENTS

4^i(8j + 7)  1^7  1^7 == 5 (mod 8) (when i = 0), or 2 (when i = 1), or 6 (when i >= 2). Thus, each nonnegative integer can be written as a sum of three squares and two nonnegative 7th powers; i.e., a(n) > 0.
More generally, each nonnegative integer can be written as a sum of three squares and a nonnegative kth power and a nonnegative mth power.


LINKS



EXAMPLE

7 = 0^2 + 1^2 + 2^2 + 1^7 + 1^7 = 1^2 + 1^1 + 2^2 + 0^7 + 1^7, a(7) = 2.
10 = 0^2 + 0^2 + 3^2 + 0^7 + 1^7 = 0^2 + 1^1 + 3^2 + 0^7 + 0^7 = 0^2 + 2^2 + 2^2 + 1^7 + 1^7 = 1^2 + 2^1 + 2^2 + 0^7 + 1^7, a(10) = 4.


MATHEMATICA

a[n_]:=Sum[If[x^2+y^2+z^2+u^7+v^7==n, 1, 0], {x, 0, n^(1/2)}, {y, x, (nx^2)^(1/2)}, {z, y, (nx^2y^2)^(1/2)}, {u, 0, (nx^2y^2z^2)^(1/7)}, {v, u, (nx^2y^2z^2u^7)^(1/7)}]
Table[a[n], {n, 0, 86}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



