%I #38 Jan 27 2023 20:01:35
%S 1715,6860,12635,15435,27440,42875,47915,50540,53235,61740,84035,
%T 109760,113715,138915,171500,191660,202160,207515,212940,218435,
%U 246960,289835,302715,315875,329315,336140,385875,415835,431235,439040,454860,479115,495635,555660,582435,619115,686000
%N Numbers which are sum of three squares of positive numbers and also 5 times of the sum of their joint products.
%C The general problem is to find such numbers which 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).
%C For k=1 it is simply a(n) = 3*n^2 given by A033428.
%C For k=2 it is A347360.
%C The present sequence is for the next k=5.
%C All possible k-numbers are listed by A331605.
%D E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985.
%e a(n) ( x, y, z)
%e ------ -------------
%e 1715 ( 3, 5, 41)
%e 6860 ( 6, 10, 82)
%e 12635 ( 5, 17, 111)
%e 15435 ( 9, 15, 123)
%e 27440 (12, 20, 164)
%e 42875 (15, 25, 205)
%e 47915 ( 3, 41, 215)
%e 50540 (10, 34, 222)
%e 53235 ( 5, 41, 227)
%e 61740 (18, 30, 246)
%e 84035 (21, 35, 287)
%e 109760 (24, 40, 328)
%Y Cf. A000378, A033428, A331605 (all possible k-numbers), A347360.
%K nonn
%O 1,1
%A _Alexander Kritov_, Sep 23 2021