A349181 Powerful harmonic numbers: numbers k such that the set of powerful divisors of k that are larger than 1 has more than one element and that the harmonic mean of this set is an integer.

100, 300, 700, 1100, 1225, 1300, 1700, 1900, 2100, 2300, 2450, 2900, 3100, 3300, 3675, 3700, 3900, 4100, 4225, 4300, 4700, 5100, 5300, 5700, 5900, 6100, 6700, 6900, 7100, 7300, 7350, 7700, 7900, 8300, 8450, 8700, 8900, 9100, 9300, 9700, 10100, 10300, 10700, 10900

Offset: 1

Author

Amiram Eldar, Nov 09 2021

Keywords

• nonn

Comments

Numbers with a single powerful divisor > 1 are A060687 and trivially have an integer harmonic mean.

The least term that is not divisible by 5 (or 25) is a(5446) = 1413721.

Examples

			100 is a term since its powerful divisors > 1 are 4, 25 and 100 and their harmonic mean, 10, is an integer.
		

Links

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

Crossrefs

Cf. A001694, A060687.

Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

Programs

Mathematica

powQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 1; powHarmQ[n_] := Module[{d = Select[Divisors[n], powQ]}, Length[d] > 1 && IntegerQ[HarmonicMean[d]]]; Select[Range[10^4], powHarmQ]