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

Author

S. I. Dimitrov, Jun 19 2025

Keywords

• nonn
• hard

Comments

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).

Examples

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

Links

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

Crossrefs

Cf. A000203, A125492, A384814, A384487.

Programs

PARI

\\ See Corneth link

Extensions

Corrected and extended by David A. Corneth, Jun 20 2025