A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

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

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 k-th power and a nonnegative m-th power.

Links

Table of n, a(n) for n=0..86.

Wikipedia, Legendre's three-square theorem

Wikipedia, Lagrange's four-square theorem

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, (n-x^2)^(1/2)}, {z, y, (n-x^2-y^2)^(1/2)}, {u, 0, (n-x^2-y^2-z^2)^(1/7)}, {v, u, (n-x^2-y^2-z^2-u^7)^(1/7)}]
Table[a[n], {n, 0, 86}]

Crossrefs

Cf. A000164, A002635, A004215, A004771, A297970.

Sequence in context: A022923 A276858 A247656 * A192454 A340944 A270533

Adjacent sequences: A294078 A294079 A294080 * A294082 A294083 A294084

Keyword

nonn

Author

XU Pingya, Feb 09 2018

Status

approved