A388012 Numbers k such that sigma(k)/k >= 5/3.

4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162, 164, 165, 168, 170, 172, 174, 176

Offset

1,1

Comments

It is not known whether sigma(k)/k = 5/3 for any k, so conjecturally, these are all numbers k such that sigma(k)/k > 5/3. However, if such a k exists, then it must by necessity be an odd square that is a multiple of 3, but not a multiple of 5 (and thus in A388016). Furthermore, 5*k is then an odd perfect number. See theorem 3 in Weiner paper.

Note that abundancy ratio 127/63 has similar properties, as it is an abundancy ratio of an odd square divisible by 3, but here we know that such an odd square exists, being 173369889 = (3*3*7*11*19)^2. - Peter Munn and Antti Karttunen, Sep 25 2025

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 4, 39, 426, 4196, 42043, 419385, 4199351, 41981198, 419755905, 4197920239, ... . Apparently, the asymptotic density of this sequence exists and equals 0.419... . - Amiram Eldar, Nov 10 2025

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

C. A. Holdener and J. A. Holdener, Characterizing Quasi-Friendly Divisors, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.4.

Paul A. Weiner, The abundancy ratio, a measure of perfection, Math. Mag. 73 (4) (2000) 307-310. (See Theorem 3)

Index entries for sequences related to sigma(n).

Mathematica

A388012Q[k_] := 3*DivisorSigma[1, k] >= 5*k;
Select[Range[200], A388012Q] (* Paolo Xausa, Sep 13 2025 *)

Prog

(PARI) is_A388012(n) = ((sigma(n)/n)>=(5/3));

Crossrefs

Positions of positive terms in A388013.

Cf. A000203, A388011 (complement).

Subsequence of A353537.

Subsequences: A388014 (primitive terms), A388016 (odd squares that are multiples of 3, but not multiples of 5), A388017.

Sequence in context: A210939 A175246 A353537 * A134928 A377848 A279040

Adjacent sequences: A388009 A388010 A388011 * A388013 A388014 A388015

Keyword

nonn,easy

Author

Antti Karttunen, Sep 13 2025

Status

approved