A307820 The number of infinitary abundant numbers below 10^n.

0, 12, 114, 1270, 12518, 125634, 1257749, 12570993, 125716733, 1256921422, 12570417639

Offset: 1

Author

Amiram Eldar, Apr 30 2019

Keywords

• nonn
• more

Examples

			Below 10^2 there are 12 infinitary abundant numbers, 24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, and 96, thus a(2) = 12.
		

Crossrefs

Cf. A049417, A129656, A302992, A302993, A302994, A307821, A307823.

Programs

Mathematica

fun[p_, e_] := Module[{ b = IntegerDigits[e, 2]}, m=Length[b]; Product[If[b[[j]] > 0, 1+p^(2^(m-j)), 1], {j, 1, m}]]; isigma[1]=1; isigma[n_] := Times @@ fun @@@ FactorInteger[n];  c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ isigma[k]>2k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq

Formula

Conjecture: Lim_{n->oo} a(n)/10^n = 0.125... is the density of infinitary abundant numbers.

Extensions

a(11) from Amiram Eldar, Sep 09 2022