A335832 Numbers k with record values of the ratio d(k)/id(k) between the number of divisors and the number of infinitary divisors.

1, 4, 16, 144, 256, 1296, 2304, 20736, 518400, 1679616, 5308416, 12960000, 41990400, 132710400, 429981696, 635040000, 1049760000, 3317760000, 10749542400, 31116960000, 51438240000, 162570240000, 268738560000, 2520473760000, 7965941760000, 13168189440000, 167961600000000

Offset: 1

Author

Amiram Eldar, Jun 25 2020

Keywords

• nonn

Comments

This sequence is infinite since the ratio d(k)/id(k) is unbounded. For example, for k = 2^(2^m) we have d(k)/id(k) = (2^m+1)/2.

The corresponding record values are 1, 1.5, 2.5, 3.75, 4.5, 6.25, 6.75, 11.25, 16.875, 20.25, ...

Links

• Amiram Eldar, Table of n, a(n) for n = 1..96

Crossrefs

Subsequence of A025487.

Cf. A000005, A037445, A307870.

Programs

Mathematica

id[1] = 1; id[n_] := Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[All, 2]]]; f[1] = 1; f[n_] := DivisorSigma[0, n]/id[n]; seq = {}; fm = 0; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

Formula

The ratios d(k)/id(k) for k = 1, 2, 3 and 4 are 1, 1, 1 and 3/2. The record values occur at 1 and 4.