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Numbers which are sum of three squares of positive numbers and also 5 times of the sum of their joint products.
1

%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