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

1,1

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.

Links

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

Example

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

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]

Crossrefs

Cf. A001694, A060687.

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

Sequence in context: A104908 A109633 A124437 * A138668 A017174 A202334

Adjacent sequences: A349178 A349179 A349180 * A349182 A349183 A349184

Keyword

nonn

Author

Amiram Eldar, Nov 09 2021

Status

approved