cp's OEIS Frontend

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

%I A384659 #7 Jun 07 2025 08:18:58
%S A384659 1,2,4,8,12,16,24,48,72,96,144,288,432,576,720,1440,2160,2880,4320,
%T A384659 8640,17280,21600,25920,30240,43200,60480,120960,151200,181440,241920,
%U A384659 302400,604800,907200,1209600,1814400,3326400,3628800,5443200,6350400,6652800,9979200
%N A384659 Numbers k such that A384655(k)/k > A384655(m)/m for all m < k.
%C A384659 All the terms above 2 are nonsquarefree (A013929).
%H A384659 Amiram Eldar, <a href="/A384659/b384659.txt">Table of n, a(n) for n = 1..355</a>
%F A384659 Limit_{n->oo} A384655(a(n))/a(n) = c, where c is Niven's constant (A033150).
%e A384659 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.
%t A384659 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;
%t A384659 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]
%o A384659 (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);
%o A384659 list(lim) = {my(rm = -1, r1); for(k = 1, lim, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", ")));}
%Y A384659 Subsequence of A025487.
%Y A384659 Cf. A013929, A033150, A384655.
%K A384659 nonn
%O A384659 1,2
%A A384659 _Amiram Eldar_, Jun 06 2025