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%I A223731 #19 Mar 20 2024 09:40:06
%S A223731 3,6,9,11,14,17,18,19,21,22,26,27,29,30,33,34,35,38,41,42,43,45,46,49,
%T A223731 50,51,53,54,57,59,61,62,65,66,67,69,70,73,74,75,77,78,81,82,83,86,89,
%U A223731 90,91,93,94,97,98,99,101,102,105,106,107,109,110,113,114,115,117,118
%N A223731 All positive numbers that are primitive sums of three nonzero squares.
%C A223731 These are the ordered numbers for which A223730 is not zero. The multiplicity for the number a(n) is A223730(a(n)).
%C A223731 According to the Halter-Koch reference the present sequence lists the ordered positive integers satisfying i) n not 0, 4, or 7 (mod 8) (see p.10, formula for r_3(n) attributed to A. Schinzel) and ii) n not from the set {1,2,5,10,13,25,37,58,85,130} with possibly one more positive integer member of this set which has to be >= 5*10^10 (if it exists at all). (Korollar 1. (b), p. 13). For this set see also A051952.
%C A223731 The first members with multiplicity 1 (precisely one representation) are 3, 6, 9, 11, 14, 17, 18, 19, 21, 22, 26, 27, 29, 30, 34, 35, 42, 43, 45, 46, 49, 50, 53, 61, 65, 67 ... A223732.
%C A223731 The first members with multiplicity 2 are 33, 38, 41, 51, 54, 57, 59, 62, 69, 74, 77, 81, 83, 90, 94, 98, 99, ... A223733.
%C A223731 The first members with multiplicity 3 are 66, 86, 89, 101, 110, 114, 131, 149, 153, 166, 171, 173, ... A223734.
%C A223731 For the complement see A223735.
%H A223731 Alois P. Heinz, <a href="/A223731/b223731.txt">Table of n, a(n) for n = 1..1000</a>
%H A223731 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.
%F A223731 The sequence a(n) is obtained from the ordered set
%F A223731 {m positive integer | m = a^2 + b^2 + c^2 , a,b,c integer, 0 < a <= b <= c, gcd(a,b,c) = 1} with entries appearing only once.
%F A223731 Conjectured g.f.: (x^77 +2*x^76 -2*x^75 +x^74 -x^73 -x^72 +2*x^50 -x^49 +2*x^47 -2*x^46 -x^45 +x^34 +2*x^33 -2*x^32 +x^31 -x^30 -x^29 +2*x^22 -x^21 +2*x^19 -2*x^18 -x^17 +3*x^15 -2*x^14 +x^13 -x^12 -x^10 +2*x^9 +2*x^7 +2*x^6 -3*x^4 -2*x^3 -3*x^2 -3*x -3)*x / (-x^6 +x^5 +x -1). - _Alois P. Heinz_, Apr 06 2013
%e A223731 a(12) = 27 because 27 is the 12th number for which A223730 is nonzero. Because A223730(27) = 1 there is only one primitive sum of three nonzero squares which is 27 denoted by [1,1,5]:
%e A223731 1^2 + 1^2 + 5^2 = 27.
%e A223731 a(28) = 54 has two primitive representations in question, namely [1,2,7] and [2,5,5]. A223730(54) = 2. The representation [3,3,6] is not primitive because gcd(3,3,6) = 3 not 1.
%e A223731 a(34) = 66 has three representations in question, namely [1,1,8], [1,4,7] and [4,5,5].
%t A223731 threeSquaresQ[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ] != {}; Select[Range[120], threeSquaresQ] (* _Jean-François Alcover_, Jun 21 2013 *)
%Y A223731 Cf. A223730, A000408 (non-primitive case), A223735 (complement).
%K A223731 nonn
%O A223731 1,1
%A A223731 _Wolfdieter Lang_, Apr 05 2013