A384658 -id:A384658 - cp's OEIS frontend

24, 48, 72, 80, 96, 108, 112, 120, 144, 160, 168, 180, 192, 200, 216, 224, 240, 252, 264, 280, 288, 300, 312, 320, 324, 336, 352, 360, 384, 396, 400, 408, 416, 420, 432, 440, 448, 456, 468, 480, 504, 520, 528, 540, 552, 560, 576, 600, 612, 624, 640, 648, 660, 672, 684, 696

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

All the terms are nonsquarefree (A013929) since A384655(n) = A051953(n) < n for squarefree numbers n.
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.
A384655(36) = 36. Are there any other numbers with this property? There are none below 10^10.
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... .

			24 is a term since A384655(24) = 25 > 24.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A013929.
A384658 is a subsequence.
Cf. A005117, A051953, A384655.

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]

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);}

cp's OEIS Frontend

A384657 Numbers k such that A384655(k) > k.

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