A224771 -id:A224771 - cp's OEIS frontend

A224773 One half of the even terms of A224771.

7, 13, 15, 19, 21, 23, 25, 27, 31, 33, 35, 37, 39, 43, 45, 47, 49, 53, 55, 57, 59, 61, 63, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 1

Wolfdieter Lang, Apr 19 2013

nonn

2*a(n), n >= 1, gives all even numbers representable as a primitive sum of three distinct nonzero squares.

The numbers a(n) are of interest in a problem related to integer solutions for the curvatures of the Descartes-Steiner circle theorem.

			a(1) = 7 because the first even term of A224771 is 14 (m = 1).
a(9) = 31 because 62 is the ninth even term of A224771 (m = 18).

Eric Weisstein's World of Mathematics, Descartes Circle Theorem.

Cf. A224771.

a(n) is one half of the n-th even term of A224771.

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

Original entry on oeis.org

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

Wolfdieter Lang, Apr 19 2013

nonn

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.

			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).

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

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

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 *)

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.

cp's OEIS Frontend

A224773 One half of the even terms of A224771.

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