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A224448 Nonnegative numbers that have a representation as a sum of three primitive and distinct squares (square 0 allowed).

%I A224448 #17 Mar 20 2024 09:40:12
%S A224448 5,10,13,14,17,21,25,26,29,30,34,35,37,38,41,42,45,46,49,50,53,54,58,
%T A224448 59,61,62,65,66,69,70,73,74,75,77,78,81,82,83,85,86,89,90,91,93,94,97,
%U A224448 98,101,105,106,107,109,110,113,114,115,117,118,121,122,125,126,129,130
%N A224448 Nonnegative numbers that have a representation as a sum of three primitive and distinct squares (square 0 allowed).
%C A224448 These are the numbers a(n) satisfying A224447(a(n)) = k >= 1, and k gives their multiplicity. See the comments on A224447 for more details and a F. Halter-Koch corollary (Korollar 1. (c), p. 13 with the first line of r_3(n) on p. 11) according to which this sequence gives the increasingly ordered numbers satisfying: neither congruent 0, 4, 7 (mod 8) nor a member 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, ?}, with a number $ >= 5*10^10 if it exists at all.
%H A224448 Vincenzo Librandi, <a href="/A224448/b224448.txt">Table of n, a(n) for n = 1..1000</a>
%H A224448 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 A224448 a(n) is the n-th largest number m satisfying m = a^2 + b^2 + c^2, with a, b, and c integers, 0 <= a < b < c, and gcd(a,b,c) = 1.
%F A224448 a(n) is the n-th largest number m for which A224447(m) > 0.
%e A224448 Denote a representation in question by the triple [a, b, c].
%e A224448 The representations for n= 1, 2, ..., 10 are:
%e A224448 n=1,   5: [0, 1, 2],
%e A224448 n=2,  10: [0, 1, 3],
%e A224448 n=3,  13: [0, 2, 3],
%e A224448 n=4,  14: [1, 2, 3],
%e A224448 n=5,  17: [0, 1, 4], [2, 2, 3],
%e A224448 n=6,  21: [1, 2, 4],
%e A224448 n=7,  25: [0, 0, 5], [0, 3, 4],
%e A224448 n=8,  26: [0, 1, 5], [1, 3, 4],
%e A224448 n=9,  29: [0, 2, 5], [2, 3, 4]
%e A224448 n=10, 30: [1, 2, 5].
%t A224448 representableQ[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 & ]] > 0; Select[ Range[130], representableQ] (* _Jean-François Alcover_, Apr 10 2013 *)
%Y A224448 Cf. A224447, A047449 (primitive case).
%K A224448 nonn
%O A224448 1,1
%A A224448 _Wolfdieter Lang_, Apr 09 2013