This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384657 #8 Jun 07 2025 08:18:47
%S A384657 24,48,72,80,96,108,112,120,144,160,168,180,192,200,216,224,240,252,
%T A384657 264,280,288,300,312,320,324,336,352,360,384,396,400,408,416,420,432,
%U A384657 440,448,456,468,480,504,520,528,540,552,560,576,600,612,624,640,648,660,672,684,696
%N A384657 Numbers k such that A384655(k) > k.
%C A384657 All the terms are nonsquarefree (A013929) since A384655(n) = A051953(n) < n for squarefree numbers n.
%C A384657 If k is a term then any positive multiple of k is also a term (since A384655(m*k) >= m * A384655(k) for any m >= 1). The primitive terms are in A384658.
%C A384657 A384655(36) = 36. Are there any other numbers with this property? There are none below 10^10.
%C A384657 The numbers of terms that do not exceed 10^k, for k = 2, 3, ..., are , 5, 80, 800, 8093, 80201, 803227, 8040424, 80374866, 803561953, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0803... .
%H A384657 Amiram Eldar, <a href="/A384657/b384657.txt">Table of n, a(n) for n = 1..10000</a>
%e A384657 24 is a term since A384655(24) = 25 > 24.
%t A384657 f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; q[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; If[emax < 2, False, s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s > n]]; Select[Range[700], q]
%o A384657 (PARI) isok(m) = {my(f = factor(m), p, e, emax, s); if(issquarefree(f), 0, p = f[,1]; e = f[,2]; emax = vecmax(e); s = emax*m; for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, p[i]^(e[i]-k)))); s > m);}
%Y A384657 Subsequence of A013929.
%Y A384657 A384658 is a subsequence.
%Y A384657 Cf. A005117, A051953, A384655.
%K A384657 nonn
%O A384657 1,1
%A A384657 _Amiram Eldar_, Jun 06 2025