A385749 Numbers z such that there exist two integers 0<x<=y<=z such that (x^2/sigma(x)^2 + y^2/sigma(y)^2 + z^2/sigma(z)^2) * (x + y + z)^2 = x^2 + y^2 + z^2.

120, 672, 1740, 2556, 4680, 11556, 27312, 32136, 41412, 41952, 42168

Offset

1,1

Comments

The numbers x, y and z form a WHM(2)-amicable triple (WHM = weighted harmonic mean). An amicable triple forms a WHM(2)-amicable triple, so the larger member of an amicable triple A125492 is a term of this sequence.

Links

Table of n, a(n) for n=1..11.

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Example

(1980, 2016, 2556) is such a triple because (1980^2/sigma(1980)^2 + 2016^2/sigma(2016)^2 + 2556^2/sigma(2556)^2)*(1980 + 2016 + 2556)^2 = 1980^2 + 2016^2 + 2556^2.
Other kinds of triples are: (120,120,120), (1560, 1740, 1740) and (117, 117, 4680).
Note that (117, 117, 4680) is the only known WHM(2)-amicable triple that is not an amicable triple.

Crossrefs

Cf. A000203, A125492, A384487.

Cf. A005820 (a subsequence, for (x,x,x) triples).

Sequence in context: A388026 A342923 A386010 * A292365 A388025 A306373

Adjacent sequences: A385746 A385747 A385748 * A385750 A385751 A385752

Keyword

nonn,hard,more

Author

S. I. Dimitrov, Jul 08 2025

Status

approved