A348826 Numbers k such that the denominator of the harmonic mean of the divisors of k is larger than 2*k.

36, 100, 144, 324, 400, 576, 900, 1296, 1600, 1936, 2304, 2500, 3600, 4356, 4624, 5184, 6400, 8100, 8464, 9216, 10000, 10404, 11664, 12100, 13456, 14400, 17424, 18496, 19044, 20736, 22500, 25600, 26244, 28900, 30276, 30976, 32400, 38416, 40000, 41616, 46656, 48400

Offset: 1

Author

Amiram Eldar, Nov 01 2021

Keywords

• nonn

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.

Examples

			36 is a term since the harmonic mean of the divisors of 36 is 324/91 and 91 > 2*36 = 72.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A099377, A099378.

Subsequence of A005101 and A348825.

A348827 is a subsequence.

Programs

Mathematica

q[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]] > 2*n; Select[Range[50000], q]

PARI

isok(k) = my(d=divisors(k)); (denominator(#d/sum(i=1, #d, 1/d[i])) > 2*k); \\ Michel Marcus, Nov 01 2021