A224772 Multiplicities for representations of numbers as primitive sums of three distinct nonzero squares.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 0, 1, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 2, 1, 1, 0, 1, 3, 0, 0, 1, 1, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 1, 3, 0, 0, 1, 2, 1, 0, 2

Offset

1,62

Comments

a(n) = k, for n >= 1, if there are exactly k representations of n as a primitive sum of three distinct nonzero squares. If a(n) = 0 then n has no such representation.

The increasingly ordered numbers with a(n) > 0 are given in A224771.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Formula

a(n) = k if n = a^2 + b^2 + c^2, a, b, and c integers, 0 < a < b < c and gcd(a,b,c) = 1, for exactly k different triples (a, b, c). If there is no such triple then a(n) = 0.

Example

a(14) = 1 because the first number with a representation in question, denoted by a triple (a, b, c), is 14, with the unique triple (1, 2, 3).
a(62) = 2 for the first number 62 which has two representations, denoted by (1, 5, 6) and (2, 3, 7).
a(101) = 3 for the first number 101 with three triples, namely (1, 6, 8), (2, 4, 9) and (4, 6, 7).

Mathematica

nn = 10; t = Table[0, {nn^2}]; Do[If[GCD[a, b, c] == 1, n = a^2 + b^2 + c^2; If[n <= nn^2, t[[n]]++]], {a, nn}, {b, a + 1, nn}, {c, b + 1, nn}]; t (* T. D. Noe, Apr 20 2013 *)

Crossrefs

Cf. A224771, A025442 (multiplicities for sums of three distinct nonzero squares).

Sequence in context: A281457 A159708 A144625 * A025442 A260118 A128582

Adjacent sequences: A224769 A224770 A224771 * A224773 A224774 A224775

Keyword

nonn

Author

Wolfdieter Lang, Apr 19 2013

Status

approved