A348412 Numbers whose even divisors have an integer harmonic mean.

2, 6, 12, 30, 56, 84, 168, 270, 280, 540, 616, 840, 992, 1092, 1344, 2856, 2976, 3276, 3780, 4590, 5320, 5940, 7560, 12400, 12420, 14880, 16256, 16380, 18848, 24360, 26784, 36036, 37200, 37240, 41664, 48768, 49140, 55692, 60480, 65520, 86304, 86800, 111720, 128520

Offset: 1

Author

Amiram Eldar, Oct 17 2021

Keywords

• nonn

Comments

The corresponding harmonic means are 2, 3, 4, 5, 6, 7, 9, 9, 10, 12, 11, 15, 10, 13, 16, 17, 15, ...

Equivalently, even numbers k such that the harmonic mean of the divisors of k/2 is either an integer (A001599) or a half-integer (A348411).

Examples

			6 is a term since its even divisors are 2 and 6, and their harmonic mean, 1/((1/2 + 1/6)/2) = 3, is an integer.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..569

Crossrefs

Cf. A001599, A074400, A348411.

A139256 is a subsequence.

Programs

Mathematica

Select[Range[2, 10^5, 2], IntegerQ[HarmonicMean[Select[Divisors[#], EvenQ]]] &]

PARI

isok(m) = if (! (m%2), my(d=select(x->!(x%2), divisors(m))); denominator(#d/sum(k=1, #d, 1/d[k])) == 1); \\ Michel Marcus, Oct 31 2021

Python

from sympy import gcd, divisor_sigma
A348412_list = [2*n for n in range(1,10**3) if (lambda x, y: 2*gcd(x,y*n)>=x)(divisor_sigma(n),divisor_sigma(n,0))] # Chai Wah Wu, Oct 20 2021