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

Author

Amiram Eldar, Jun 13 2020

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

Formula

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