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%I #101 Oct 04 2021 09:02:34
%S 18,72,98,162,288,338,392,450,648,722,882,1152,1352,1458,1568,1800,
%T 1922,2178,2450,2592,2738,2888,3042,3528,3698,4050,4608,4802,5202,
%U 5408,5832,6272,6498,7200,7442,7688,7938,8450,8712,8978,9522,9800,10368,10658,10952,11250,11552,11858
%N Numbers that can be represented as the sum of squares of 3 numbers and also equal to twice the sum of their joint products.
%C Integers that can be represented as the sum of three squares of integers x, y, z, and additionally also satisfy x^2+y^2+z^2 = k *(x*y+ x*z + y*z), with k=2.
%C All possible k are given by A331605.
%D E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985.
%F Empirically, such numbers appear to be a(n) = 2*b_n^2 where b_n are numbers whose product of prime indices is even (A324929).The triplet (x,y,x) is always (n*k^2, n*m^2, n*p^2).
%e For example, the third term (1,4,9) is 1^2+4^2+9^2 = 2*(1*4+1*9+4*9) = 98.
%e The sequence is given by
%e a(n) (x, y, z)
%e 18 (1,1,4)
%e 72 (2,2,8)
%e 98 (1,4,9)
%e 162 (3,3,12)
%e 288 (4,4,16)
%e 338 (1,9,16)
%e 392 (2,8,18)
%e 450 (5,5,20)
%e 648 (6,6,24)
%e 722 (4,9,25)
%e 882 (1,16,25) (3,12,27) (7,7,28)
%e 1152 (8,8,32) (2,18,32)
%e 1352 (2,18,32)
%e 1458 (9,9,36)
%e 1568 (4,16,36)
%e 1800 (10,10,40)
%e 1922 (1,25,36)
%e 2178 (11,11,44)
%e 2450 (5,20,45)
%e 2592 (12,12,48)
%e 2738 (9,16,49)
%e 2888 (8,18,50)
%e 3042 (3,27,48) (4,25,49) (13,13,52)
%e 3528 (2,32,50) (6,24,54)
%t q[n_] := (s = Select[PowersRepresentations[n,3,2], AllTrue[#, #1 > 0 &]&]) != {} && MemberQ[(#[[1]]*#[[2]] + #[[2]]*#[[3]] + #[[3]]*#[[1]])& /@ s, n/2]; Select[Range[2, 12000, 2], q] (* _Amiram Eldar_, Oct 03 2021 *)
%Y Subsequence of A000378. Complement of A004215.
%Y Cf. A033428 (case k=1), A324929, A331605 (k-numbers).
%K nonn
%O 1,1
%A _Alexander Kritov_, Sep 22 2021
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