|
%I #7 Nov 10 2021 07:07:40
%S 100,300,700,1100,1225,1300,1700,1900,2100,2300,2450,2900,3100,3300,
%T 3675,3700,3900,4100,4225,4300,4700,5100,5300,5700,5900,6100,6700,
%U 6900,7100,7300,7350,7700,7900,8300,8450,8700,8900,9100,9300,9700,10100,10300,10700,10900
%N 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.
%C Numbers with a single powerful divisor > 1 are A060687 and trivially have an integer harmonic mean.
%C The least term that is not divisible by 5 (or 25) is a(5446) = 1413721.
%H Amiram Eldar, <a href="/A349181/b349181.txt">Table of n, a(n) for n = 1..10000</a>
%e 100 is a term since its powerful divisors > 1 are 4, 25 and 100 and their harmonic mean, 10, is an integer.
%t 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]
%Y Cf. A001694, A060687.
%Y Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.
%K nonn
%O 1,1
%A _Amiram Eldar_, Nov 09 2021
|
