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.

18, 72, 98, 162, 288, 338, 392, 450, 648, 722, 882, 1152, 1352, 1458, 1568, 1800, 1922, 2178, 2450, 2592, 2738, 2888, 3042, 3528, 3698, 4050, 4608, 4802, 5202, 5408, 5832, 6272, 6498, 7200, 7442, 7688, 7938, 8450, 8712, 8978, 9522, 9800, 10368, 10658, 10952, 11250, 11552, 11858

Offset: 1

Author

Alexander Kritov, Sep 22 2021

Keywords

• nonn

Comments

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.

All possible k are given by A331605.

Examples

			For example, the third term (1,4,9) is 1^2+4^2+9^2 = 2*(1*4+1*9+4*9) = 98.
The sequence is given by
   a(n)    (x, y, z)
    18     (1,1,4)
    72     (2,2,8)
    98     (1,4,9)
   162     (3,3,12)
   288     (4,4,16)
   338     (1,9,16)
   392     (2,8,18)
   450     (5,5,20)
   648     (6,6,24)
   722     (4,9,25)
   882     (1,16,25) (3,12,27)  (7,7,28)
  1152     (8,8,32)  (2,18,32)
  1352     (2,18,32)
  1458     (9,9,36)
  1568     (4,16,36)
  1800     (10,10,40)
  1922     (1,25,36)
  2178     (11,11,44)
  2450     (5,20,45)
  2592     (12,12,48)
  2738     (9,16,49)
  2888     (8,18,50)
  3042     (3,27,48) (4,25,49) (13,13,52)
  3528     (2,32,50) (6,24,54)
		

References

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

Crossrefs

Subsequence of A000378. Complement of A004215.

Cf. A033428 (case k=1), A324929, A331605 (k-numbers).

Programs

Mathematica

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 *)

Formula

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).