A384659 Numbers k such that A384655(k)/k > A384655(m)/m for all m < k.

1, 2, 4, 8, 12, 16, 24, 48, 72, 96, 144, 288, 432, 576, 720, 1440, 2160, 2880, 4320, 8640, 17280, 21600, 25920, 30240, 43200, 60480, 120960, 151200, 181440, 241920, 302400, 604800, 907200, 1209600, 1814400, 3326400, 3628800, 5443200, 6350400, 6652800, 9979200

Offset: 1

Author

Amiram Eldar, Jun 06 2025

Keywords

• nonn

Comments

All the terms above 2 are nonsquarefree (A013929).

Examples

			The first values of A384655(k)/k, for k = 1..8, are {0, 1/2, 1/3, 3/4, 1/5, 2/3, 1/7, 7/8}. The record values, 0, 1/2, 3/4 and 7/8, occur at k = 1, 2, 4 and 8, the first 4 terms of this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..355

Crossrefs

Subsequence of A025487.

Cf. A013929, A033150, A384655.

Programs

Mathematica

f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; r[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s/n]; r[1] = 0;
seq[lim_] := Module[{s = {}, rm = -1, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]

PARI

r(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, p[i]^(e[i]-k)))); s/n);
list(lim) = {my(rm = -1, r1); for(k = 1, lim, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", ")));}

Formula

Limit_{n->oo} A384655(a(n))/a(n) = c, where c is Niven's constant (A033150).