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%I #9 Aug 20 2015 06:39:08
%S 30,39,46,50,51,54,57,62,63,65,66,70,71,74,75,78,79,81,84,85,86,87,90,
%T 91,93,94,95,98,99,102,105,106,107,109,110,111,113,114,116,117,118,
%U 119,121,122,123,125,126,127,129,130,131,133,134,135,137,138,139,140
%N Numbers which are a sum of four distinct nonzero squares where the summands have no common factor > 1.
%C A primitive representation of a number m as a sum of four distinct nonzero squares is determined from a quadruple [s(1), s(2), s(3), s(4)] of integers with 0 < s(1) < s(2) < s(3) < s(4) with gcd(s(1),s(2),s(3),s(4)) = 1, and m = sum(s(j)^2, j=1..4). If m has such a primitive representation then k^2*m, with integer k > 0, has trivially a non-primitive representation. Therefore primitive representations are of interest.
%C For the multiplicities see A223728.
%C This sequence is a proper subset of A004433. The first entry of A004433 missing here is 120 = A004433(43). The first common entry with different multiplicity is A004433(72) = 156 = a(71) with two primitive representations with quadruples
%C [1, 3, 5, 11] and [1, 5, 7, 9]. [2, 4, 6, 10] = 2*[1, 2, 3, 5]is a non-primitive representation due to 156 = 4*39.
%F This sequence are the increasingly ordered members of the set {m an integer | m = sum(s(j)^2, j=1..4), with 0 < s(1) < s(2) < s(3) < s(4) and gcd(s(1),s(2),s(3),s(4)) = 1}.
%e a(1) = 30 because the numbers 0,...,29 have no representation as a sum of four distinct nonzero squares, and 30 has one representation given by the quadruple [1,2,3,4] which is primitive.
%e a(16) = 78 has three such representations given by the quadruples [1, 2, 3, 8], [1, 4, 5, 6] and [2, 3, 4, 7] which are all primitive. Hence A223728(16) = 3. This is the first entry with more than one (primitive) representation.
%e a(23) = 90 has multiplicity 2 = A223728 because there are two primitive quadruples [1, 2, 6, 7] and [1, 3, 4, 8].
%e a(71) = 156 has multiplicity A223728(71) = 2 (see a comment above).
%Y Cf. A222949, A097203, A223728, A259058 (multiplicity >= 2 instances).
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Mar 27 2013
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