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.
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
KEYWORD
nonn
AUTHOR
Alexander Kritov, Sep 22 2021
STATUS
approved