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

%I A224447 #22 Mar 24 2020 15:24:10
%S A224447 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,
%T A224447 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,
%U A224447 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
%N A224447 Multiplicities for representations of n as primitive and distinct sums of three squares.
%C A224447 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.
%C A224447 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.
%C A224447 See A224444 for the multiplicities for primitive sums of three squares (square 0 allowed).
%C A224447 The numbers for which a(n) is not 0 are given in A224448.
%H A224447 F. Halter-Koch, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf">Darstellung natürlicher Zahlen als Summe von Quadraten</a>, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.
%F A224447 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.
%e A224447 Denote a representation in question by an increasingly ordered triple [a, b, c].
%e A224447 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.
%e A224447 a(6) = 0 because the only primitive representation (A224444(6) = 1) is [1, 1, 2], but the entries are not distinct.
%e A224447 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.
%e A224447 a(26) = 2 with the primitive representations (A224444(26) = 2) given by [0, 1, 5] and [1, 3, 4] which both have distinct entries.
%t A224447 a[n_] := Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 &] // Length; Table[a[n], {n, 0, 130}] (* _Jean-François Alcover_, Apr 10 2013 *)
%Y A224447 Cf. A224444 (primitive case), A224448.
%K A224447 nonn
%O A224447 0,27
%A A224447 _Wolfdieter Lang_, Apr 09 2013