A335541 Numbers with a record value of the ratio of the number of abundant divisors to the total number of divisors.

1, 12, 24, 36, 72, 120, 144, 216, 360, 432, 720, 1440, 2160, 2880, 4320, 8640, 12960, 17280, 20160, 25920, 30240, 40320, 51840, 60480, 80640, 120960, 181440, 241920, 362880, 483840, 604800, 725760, 967680, 1209600, 1451520, 1814400, 2177280, 2419200, 2903040, 3628800

Offset

1,2

Comments

Apparently, all the terms are least numbers of their prime signature (A025487). This was verified for the first 78 terms.

The ratio A080224(m)/A000005(m) can be arbitrarily close to 1. For example, A080224(6^k)/A000005(6^k) = (k-1)/(k+1) = 1 - 2/(k+1), for k >= 1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..78

Formula

Numbers m such that A080224(m)/A000005(m) > A080224(k)/A000005(k) for all k < m.

Example

36 has 9 divisors, {1, 2, 3, 4, 6, 9, 12, 18, 36}, 3 of which are abundant, {12, 18, 36}. The ratio 3/9 = 1/3 is larger than the ratios for all the numbers below 36. Hence 36 is a term.

Mathematica

f[n_] := Count[(d = Divisors[n]), _?(DivisorSigma[1, #] > 2# &)]/Length[d]; fm = -1; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^4}]; s

Crossrefs

Cf. A000005, A000203, A005100, A025487, A080224, A335540.

Sequence in context: A022759 A335540 A091193 * A174018 A098242 A139406

Adjacent sequences: A335538 A335539 A335540 * A335542 A335543 A335544

Keyword

nonn

Author

Amiram Eldar, Jun 13 2020

Status

approved