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%I #17 Jul 18 2024 09:18:47
%S 1,2,6,30,210,2310,30030,480480,510510,8168160,9699690,155195040,
%T 223092870,3569485920,6469693230,103515091680,200560490130,
%U 3208967842080,7420738134810,118731810156960,304250263527210,4868004216435360,13082761331670030,209324181306720480
%N 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.
%C 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.
%C The corresponding record values are 1, 2, 4, 8, 16, 32, 64, 96, 128, ...
%H Amiram Eldar, <a href="/A361807/b361807.txt">Table of n, a(n) for n = 1..84</a>
%e 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.
%t 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]
%o (PARI) r(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 2]+1)/numdiv(f[i, 2])); }
%o lista(kmax) = {my(rm = 0, r1); for(k = 1, kmax, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }
%Y Subsequence of A025487.
%Y Cf. A000005, A002110, A049419.
%Y Similar sequences: A307870, A335832.
%K nonn
%O 1,2
%A _Amiram Eldar_, Mar 25 2023
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