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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..37.

Example

    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)

Crossrefs

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

Sequence in context: A184090 A297560 A157247 * A267201 A024408 A267741

Adjacent sequences: A347966 A347967 A347968 * A347970 A347971 A347972

Keyword

nonn

Author

Alexander Kritov, Sep 23 2021

Status

approved