OFFSET
1,1
COMMENTS
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... .
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
24 is a term since A384655(24) = 25 > 24.
MATHEMATICA
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]
PROG
(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); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 06 2025
STATUS
approved
