A384487 -id:A384487 - cp's OEIS frontend

A385155 Numbers z such that there exist two integers 0

1380, 1540, 1560, 1638, 2016, 2250, 2520, 2556, 2700, 2772, 3024, 3120, 3312, 3360, 3408, 3480, 3640, 3654, 3780, 3816, 3828, 3876, 4200, 4320, 4440, 4452, 4620, 4920, 4956, 5220, 5280, 5292, 5304, 5340, 5400, 5460, 5472, 5640, 5700, 5724, 5760, 5940, 6048, 6060, 6180

Offset: 1

S. I. Dimitrov, Jun 19 2025

The numbers x, y and z form a HM(1,3)-amicable triple (HM = harmonic mean). See Dimitrov link. An amicable triple forms a HM(1,3)-amicable triple, so the larger member of an amicable triple A125492 is a term of this sequence.
From David A. Corneth, Jun 20 2025: (Start)
Let sx = sigma(x), sy = sigma(y) and sz = sigma(z).
We may write (1/sx + 1/sy + 1/sz)*(x + y + z) = (1/sx + 1/sy) * (x + y) + 1/sz * (x + y + z) + z * (1/sx + 1/sy). As z > y > x we have 2 * z > x + y so z > (x + y) / 2.
Therefore we have 3 = (1/sx + 1/sy) * (x + y) + 1/sz * (x + y + z) + z * (1/sx + 1/sy) > 1.5*(1/sx + 1/sy) * (x + y) + 1/sz * (x + y + z) > 1.5*(1/sx + 1/sy) * (x + y) and so (1/sx + 1/sy) * (x + y) < 2. Possibly 2 could be tightened due to 1/sz * (x + y + z) which is discarded for now.
If we know (1/sx + 1/sy) * (x + y) < U for some U then similarly x/s(x) can be tightened to 0.5*U and maybe more due to term 1/sy * (x + y).
Furthermore 3 = (1/sx + 1/sy + 1/sz)*(x + y + z) > 1/sx * (x + y + z) > 1/sx * (x + y + y) = 1/sx * (x + 2*y) which constraints y and later on z once 1/sx is known.
For every pair (k, m) in {(x, y), (x, z), (y, z)} from solutions (x, y, z) where z <= 10000 we have (k + m) * (1/sigma(k) + 1/sigma(m)) <= 1.5. Is that the case for every solution? (End).

			(840, 1020, 1380) is such a triple because (1/sigma(840) + 1/sigma(1020) + 1/sigma(1380))*(840 + 1020 + 1380) = 3.

David A. Corneth, PARI program
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A125492, A384814, A384487.

PARI
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\\ See Corneth link
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Corrected and extended by David A. Corneth, Jun 20 2025

A385749 Numbers z such that there exist two integers 0

Original entry on oeis.org

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

Offset: 1

S. I. Dimitrov, Jul 08 2025

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.

			(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.

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

Cf. A000203, A125492, A384487.

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

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A385155 Numbers z such that there exist two integers 0

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