OFFSET
1,1
COMMENTS
A disjoint union of the harmonic numbers (A001599) and the composite numbers whose harmonic mean of divisors is of the form m - 1/k, where m and k are integers.
If p is an odd prime, then the harmonic mean of its divisors is p*tau(p)/sigma(p) = p*A000005(p)/A000203(p) = p/((p+1)/2), so A099378(p) | (A099377(p) + 1). Therefore, this sequence is restricted to composite numbers.
This sequence is infinite. For example, it includes all the semiprimes of the form 3*p, where p == 2 (mod 3).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
15 is a term since it is composite, the harmonic mean of divisors of 15 is 5/2 and 2 | (5+1).
MATHEMATICA
h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := Divisible[Numerator[(h1 = h[n])] + 1, Denominator[h1]]; Select[Range[1000], CompositeQ[#] && q[#] &]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 02 2021
STATUS
approved