A363170 The number of powerful abundant numbers (A363169) not exceeding 10^n.

0, 3, 23, 82, 297, 1000, 3268, 10534, 33799, 107901, 343155, 1090189, 3460380, 10970774, 34749182, 109991778, 348006756, 1101058505, 3483105232, 11017518803

Offset: 1

Author

Amiram Eldar, May 19 2023

Keywords

• nonn
• more

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... .

Examples

			a(2) = 3 since there are 3 powerful abundant numbers not exceeding 10^2: 36, 72 and 100.
		

Crossrefs

Cf. A001694, A005101, A118896, A302992, A363169.

Programs

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]

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++); }