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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 19 01:31 EDT 2024. Contains 374388 sequences. (Running on oeis4.)