A224448 -id:A224448 - cp's OEIS frontend

A224447 Multiplicities for representations of n as primitive and distinct sums of three squares.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 3, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 2, 1, 0, 0, 1, 1, 1, 0, 2, 2, 0, 0, 3, 2, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 4, 0, 0, 0, 2, 2, 1, 0, 2, 3, 0, 0, 2, 1, 1, 0, 2, 1, 0, 0, 1, 3, 0, 0, 3, 2, 0, 0, 2, 2

Offset: 0

Wolfdieter Lang, Apr 09 2013

nonn

a(n), n >= 1 gives the number of different representations of the positive integer n as a sum of three distinct squares (square 0 allowed) which have no common factor > 1. a(0) = 0. Neither the order of the summands nor the signs of the numbers to be squared are taken into account. If a(n) = 0 there is no such representation for n.
According to a corollary by F. Halter-Koch (Korollar 1. (c), p. 13, together with the first line of r_3(n) on p. 11) a(n) > 0 if and only if n is neither congruent 0, 4, 7 (mod 8) nor an element of the set S := {1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627, ?}, and the number ? >= 5*10^10 if it exists at all. This set appears as A224449.
See A224444 for the multiplicities for primitive sums of three squares (square 0 allowed).
The numbers for which a(n) is not 0 are given in A224448.

			Denote a representation in question by an increasingly ordered triple [a, b, c].
The first nonnegative integer with a representation in question is n = 5 with a(5) = 1 because 5 has only one primitive representation (see A224444(5) = 1), namely [0, 1, 2] and the entries are distinct.
a(6) = 0 because the only primitive representation (A224444(6) = 1) is [1, 1, 2], but the entries are not distinct.
a(17) = 1 with the unique representation [0, 1, 4]. The primitive representation [2, 2, 3] (A224444(17) = 2) is excluded because it does not have distinct entries.
a(26) = 2 with the primitive representations (A224444(26) = 2) given by [0, 1, 5] and [1, 3, 4] which both have distinct entries.

F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.

Cf. A224444 (primitive case), A224448.

a[n_] := Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 &] // Length; Table[a[n], {n, 0, 130}] (* Jean-François Alcover, Apr 10 2013 *)

a(n) = k >= 1 if n, n >= 0, has k different representations as n = a^2 + b^2 + c^2, a, b and c integers, 0 <= a < b < c and gcd(a,b,c) = 1. If there is no such representation a(0) = 0.

A224449 A finite set of numbers relevant for the representation of numbers as primitive distinct sums of three squares (0 squared allowed).

Original entry on oeis.org

1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627

Offset: 1

Wolfdieter Lang, Apr 09 2013

This set of 23 numbers, possibly with one more number a >= 5*10^10, appears in a corollary of the Halter-Koch reference (Korollar 1.(c), p. 13 with the first line of r_3(n) on p. 11). A number is representable as a^2 + b^2 + c^2 with a,b, and c integers, 0 <= a < b < c, and gcd(a,b,c) = 1 if and only if n is not congruent 0, 4, 7 (mod 8) and not one of the numbers {a(k), k = 1 .. 23}, and, if it exists at all, a further number >= 5*10^10.
For the multiplicities of these representable numbers see A224447, and for the numbers themselves see A224448.
For a similar set of numbers relevant for sums of three nonzero squares see A051952.

F. Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.

Cf. A224447, A224448, A051952.

representableQ[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 & ]] > 0; Select[ Range[1000], Not[ representableQ[#] || MatchQ[ Mod[#, 8], 0 | 4 | 7]] &] (* Jean-François Alcover, Apr 10 2013 *)

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A224447 Multiplicities for representations of n as primitive and distinct sums of three squares.

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