login
A348412
Numbers whose even divisors have an integer harmonic mean.
2
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
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
A139256 is a subsequence.
Sequence in context: A179674 A038199 A056267 * A133996 A284573 A355475
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 17 2021
STATUS
approved