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

1,1

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.

Links

Table of n, a(n) for n=1..57.

Example

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.

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}]

Crossrefs

Cf. A362802.

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

Sequence in context: A326696 A097603 A374198 * A315754 A315755 A315756

Adjacent sequences: A362798 A362799 A362800 * A362802 A362803 A362804

Keyword

nonn

Author

Amiram Eldar, May 04 2023

Status

approved