cp's OEIS Frontend

A347360 Numbers that can be represented as the sum of squares of 3 numbers and also equal to twice the sum of their joint products.

%I A347360 #101 Oct 04 2021 09:02:34
%S A347360 18,72,98,162,288,338,392,450,648,722,882,1152,1352,1458,1568,1800,
%T A347360 1922,2178,2450,2592,2738,2888,3042,3528,3698,4050,4608,4802,5202,
%U A347360 5408,5832,6272,6498,7200,7442,7688,7938,8450,8712,8978,9522,9800,10368,10658,10952,11250,11552,11858
%N A347360 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 A347360 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 A347360 All possible k are given by A331605.
%D A347360 E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985.
%F A347360 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 A347360 For example, the third term (1,4,9) is 1^2+4^2+9^2 = 2*(1*4+1*9+4*9) = 98.
%e A347360 The sequence is given by
%e A347360    a(n)    (x, y, z)
%e A347360     18     (1,1,4)
%e A347360     72     (2,2,8)
%e A347360     98     (1,4,9)
%e A347360    162     (3,3,12)
%e A347360    288     (4,4,16)
%e A347360    338     (1,9,16)
%e A347360    392     (2,8,18)
%e A347360    450     (5,5,20)
%e A347360    648     (6,6,24)
%e A347360    722     (4,9,25)
%e A347360    882     (1,16,25) (3,12,27)  (7,7,28)
%e A347360   1152     (8,8,32)  (2,18,32)
%e A347360   1352     (2,18,32)
%e A347360   1458     (9,9,36)
%e A347360   1568     (4,16,36)
%e A347360   1800     (10,10,40)
%e A347360   1922     (1,25,36)
%e A347360   2178     (11,11,44)
%e A347360   2450     (5,20,45)
%e A347360   2592     (12,12,48)
%e A347360   2738     (9,16,49)
%e A347360   2888     (8,18,50)
%e A347360   3042     (3,27,48) (4,25,49) (13,13,52)
%e A347360   3528     (2,32,50) (6,24,54)
%t A347360 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 A347360 Subsequence of A000378. Complement of A004215.
%Y A347360 Cf. A033428 (case k=1), A324929, A331605 (k-numbers).
%K A347360 nonn
%O A347360 1,1
%A A347360 _Alexander Kritov_, Sep 22 2021