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

1,1

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).

Links

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

Example

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.

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A001599, A074400, A348411.

A139256 is a subsequence.

Sequence in context: A179674 A038199 A056267 * A133996 A284573 A355475

Adjacent sequences: A348409 A348410 A348411 * A348413 A348414 A348415

Keyword

nonn

Author

Amiram Eldar, Oct 17 2021

Status

approved