OFFSET
1,1
COMMENTS
Since the harmonic mean of the divisors of k is k*tau(k)/sigma(k), where tau(k) = A000005(k) and sigma(k) = A000203(k), then A099378(k) <= sigma(k). Therefore, all the terms k have sigma(k) > 2*k and are thus abundant numbers (A005101).
The first nonsquare term is a(92) = 320000. Apparently, the nonsquares are relatively rare in this sequence. For example, of the first 10^4 terms only 107 are nonsquares.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
36 is a term since the harmonic mean of the divisors of 36 is 324/91 and 91 > 2*36 = 72.
MATHEMATICA
q[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]] > 2*n; Select[Range[50000], q]
PROG
(PARI) isok(k) = my(d=divisors(k)); (denominator(#d/sum(i=1, #d, 1/d[i])) > 2*k); \\ Michel Marcus, Nov 01 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 01 2021
STATUS
approved