A349180 Coreful harmonic numbers: nonsquarefree numbers k such that the harmonic mean of the coreful divisors of k is an integer.

12, 18, 36, 56, 60, 75, 84, 90, 126, 132, 150, 156, 168, 180, 198, 204, 228, 234, 240, 252, 276, 280, 306, 342, 348, 351, 372, 392, 396, 414, 420, 444, 450, 468, 492, 504, 516, 522, 525, 558, 564, 588, 612, 616, 630, 636, 660, 666, 684, 702, 708, 720, 726, 728

Offset

1,1

Comments

A divisor of a number k is coreful if it is divisible by every prime that divides k.

The sequence is restricted to nonsquarefree numbers since the squarefree numbers have a single coreful divisor and thus they trivially have an integer harmonic mean.

Links

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

Example

12 is a term since its coreful divisors are 6 and 12 and their harmonic mean, 8, is an integer.

Mathematica

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; corHarmQ[n_] := Module[{r = rad[n], d}, d = Select[Divisors[n], rad[#] == r &]; IntegerQ[HarmonicMean[d]]]; Select[Range[10^3], !SquareFreeQ[#] && corHarmQ[#] &]

Crossrefs

Cf. A005117, A013929, A057723, A307958, A307959.

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

Sequence in context: A348965 A162694 A230354 * A197464 A124205 A338259

Adjacent sequences: A349177 A349178 A349179 * A349181 A349182 A349183

Keyword

nonn

Author

Amiram Eldar, Nov 09 2021

Status

approved