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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..48.

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).

Example

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)

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 *)

Crossrefs

Subsequence of A000378. Complement of A004215.

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

Sequence in context: A257693 A231328 A274577 * A373903 A195321 A069058

Adjacent sequences: A347357 A347358 A347359 * A347361 A347362 A347363

Keyword

nonn

Author

Alexander Kritov, Sep 22 2021

Status

approved