A361807 Numbers k with record values of the ratio A000005(k)/A049419(k) between the number of divisors of k and the number of exponential divisors of k.

1, 2, 6, 30, 210, 2310, 30030, 480480, 510510, 8168160, 9699690, 155195040, 223092870, 3569485920, 6469693230, 103515091680, 200560490130, 3208967842080, 7420738134810, 118731810156960, 304250263527210, 4868004216435360, 13082761331670030, 209324181306720480

Offset: 1

Author

Amiram Eldar, Mar 25 2023

Keywords

• nonn

Comments

This sequence is infinite since the ratio A000005(k)/A049419(k) is unbounded. For example, for k = A002110(m) we have A000005(k)/A049419(k) = 2^m.

The corresponding record values are 1, 2, 4, 8, 16, 32, 64, 96, 128, ...

Examples

			The ratios A000005(k)/A049419(k) for k = 1, 2, 3, 4, 5 and 6 are 1, 2, 2, 3/2, 2 and 4. The record values, 1, 2 and 4, occur at 1, 2 and 6, the first 3 terms of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..84

Crossrefs

Subsequence of A025487.

Cf. A000005, A002110, A049419.

Similar sequences: A307870, A335832.

Programs

Mathematica

f[p_, e_] := (e+1)/DivisorSigma[0, e]; r[1] = 1; r[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{rm = 0, s = {}, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1 , kmax}]; s]; seq[10^6]

PARI

r(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 2]+1)/numdiv(f[i, 2])); }
lista(kmax) = {my(rm = 0, r1); for(k = 1, kmax, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }