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

1,2

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, ...

Links

Table of n, a(n) for n=1..24.

Example

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.

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]

Prog

(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, ", "))); }

Crossrefs

Subsequence of A025487.

Cf. A000005, A002110, A049419.

Similar sequences: A307870, A335832.

Sequence in context: A331665 A171989 A335069 * A233438 A002110 A118491

Adjacent sequences: A361804 A361805 A361806 * A361808 A361809 A361810

Keyword

nonn

Author

Amiram Eldar, Mar 25 2023

Status

approved