OFFSET
1,2
COMMENTS
The ratios a(n)/A118896(n) seem to converge to a positive value as n grows: for n = 14..20 they are 0.506417..., 0.506728..., 0.506863..., 0.506890..., 0.506987..., 0.507059..., 0.507120... .
Conjecture: the asymptotic relative density of the abundant numbers within the powerful numbers exists and equals 0.507... .
EXAMPLE
a(2) = 3 since there are 3 powerful abundant numbers not exceeding 10^2: 36, 72 and 100.
MATHEMATICA
seq[nmax_] := Module[{c = 0, p = 10, k = 1, kmax = 10^nmax, s = {}}, While[k <= kmax, If[DivisorSigma[-1, k] > 2 && Min[FactorInteger[k][[;; , 2]]] > 1, c++]; If[k == p, AppendTo[s, c]; p *= 10]; k++]; s]; seq[5]
PROG
(PARI) is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; } \\ A363169
lista(nmax) = {my(c = 0, p = 10, k = 1, kmax = 10^nmax); while(k <= kmax, if(is(k), c++); if(k == p, print1(c, ", "); p *= 10); k++); }
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, May 19 2023
STATUS
approved