A348696 Numbers m such that there is at least one smaller number k < m with the same harmonic mean of divisors as m.

135, 224, 496, 936, 1485, 1488, 1755, 2295, 2464, 2565, 2912, 3105, 3808, 3915, 4185, 4256, 4464, 4680, 4995, 5152, 5456, 5535, 5805, 6345, 6448, 6496, 6552, 6860, 6944, 7155, 7965, 8235, 8288, 8432, 9045, 9184, 9424, 9585, 9632, 9855, 10296, 10528, 10665, 10976

Offset: 1

Author

Amiram Eldar, Oct 30 2021

Keywords

• nonn

Comments

The corresponding values of k (the least in case there are more than one) are 84, 120, 140, 864, 924, 420, 1092, 1428, 1320, 1596, ... (see the link for more values).

The least term m with more than one smaller number k with the same harmonic mean of divisors as m is m = a(1237) = A348697(1) = 321048 with k = 201096 and 296352.

Examples

			135 is a term since the harmonic mean of divisors of 135 is 9/2, and it is also the harmonic mean of divisors of 84 which is smaller than 135.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Amiram Eldar, Table of n, a(n), least k for n=1..1000

Crossrefs

Cf. A099377, A099378, A348697.

Programs

Mathematica

h = Table[DivisorSigma[0, n]/DivisorSigma[-1, n], {n, 1, 10000}]; i = Position[(t = Tally[h])[[;; , 2]], _?(# > 1 &)] // Flatten; Position[h, #][[2 ;; -1]] & /@ t[[i, 1]] // Flatten // Sort