A362801 Numbers whose set of divisors can be partitioned into disjoint parts, all of length > 1 and having integer harmonic mean.

6, 12, 18, 24, 28, 30, 40, 42, 45, 48, 54, 56, 60, 66, 72, 78, 84, 90, 96, 102, 108, 112, 114, 120, 126, 132, 135, 138, 140, 144, 150, 156, 162, 168, 174, 180, 186, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 234, 240, 246, 252, 258, 264, 270, 276

Offset: 1

Author

Amiram Eldar, May 04 2023

Keywords

• nonn

Comments

Numbers k such that A362802(k) > 0.

Includes all the harmonic numbers (A001599) except for 1, since the set of their divisors has an integer harmonic mean (in this case the partition is into a single part).

This sequence is infinite. For example, if k is a term and p is a prime that does not divide k, then k*p is also a term.

Examples

			12 is a term since its set of divisors, {1, 2, 3, 4, 6, 12} can be partitioned into 2 disjoint parts, {1, 2, 3, 6} and {4, 12}, whose harmonic means, 2 and 6, are both integers.
		

Crossrefs

Cf. A362802.

Subsequences: A001599 \ {1}, A348715, A362803 \ {1}.

Programs

Mathematica

harmQ[s_] := AllTrue[s, Length[#] > 1 && IntegerQ[HarmonicMean[#]] &]; q[n_] := Module[{d = Divisors[n], r}, r = ResourceFunction["SetPartitions"][d]; AnyTrue[r, harmQ]]; Do[If[q[n], Print[n]], {n, 1, 100}]