A347969 Numbers which are sum of three squares of positive numbers and also 5 times of the sum of their joint products.

1715, 6860, 12635, 15435, 27440, 42875, 47915, 50540, 53235, 61740, 84035, 109760, 113715, 138915, 171500, 191660, 202160, 207515, 212940, 218435, 246960, 289835, 302715, 315875, 329315, 336140, 385875, 415835, 431235, 439040, 454860, 479115, 495635, 555660, 582435, 619115, 686000

Offset: 1

Author

Alexander Kritov, Sep 23 2021

Keywords

• nonn

Comments

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).

For k=1 it is simply a(n) = 3*n^2 given by A033428.

For k=2 it is A347360.

The present sequence is for the next k=5.

All possible k-numbers are listed by A331605.

Examples

			    a(n)      ( x,  y,   z)
  ------      -------------
    1715      ( 3,  5,  41)
    6860      ( 6, 10,  82)
   12635      ( 5, 17, 111)
   15435      ( 9, 15, 123)
   27440      (12, 20, 164)
   42875      (15, 25, 205)
   47915      ( 3, 41, 215)
   50540      (10, 34, 222)
   53235      ( 5, 41, 227)
   61740      (18, 30, 246)
   84035      (21, 35, 287)
  109760      (24, 40, 328)
		

References

• E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985.

Crossrefs

Cf. A000378, A033428, A331605 (all possible k-numbers), A347360.