A307822 -id:A307822 - cp's OEIS frontend

A307821 The number of exponential abundant numbers below 10^n.

0, 0, 1, 12, 102, 1045, 10449, 104365, 1043641, 10436775, 104367354

Offset: 1

Amiram Eldar, Apr 30 2019

			Below 10^3 there is only one exponential abundant number, A129575(1) = 900, thus a(3) = 1.

Cf. A051377, A129575, A302992, A302993, A302994, A307820, A307822, A307823.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ esigma[k]>2k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq

Limit_{n->oo} a(n)/10^n = 0.001043673... is the density of exponential abundant numbers (see A129575). [Updated by Amiram Eldar, Sep 02 2022]

a(11) from Amiram Eldar, Sep 02 2022

A307961 The number of coreful perfect numbers (A307958) below 10^n.

Original entry on oeis.org

0, 1, 9, 94, 940, 9359, 93611, 936098, 9361056, 93610484, 936104756, 9361047434

Offset: 1

Amiram Eldar, May 08 2019

Cf. A307822, A307958, A307960.

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); cpQ[n_] := csigma[n] == 2*n; s={}; c=0; p=100; Do[If[k==p, AppendTo[s, c]; p*=10]; If[cpQ[k], c++], {k, 1, 1000001}]; s

a(n) ~ c * 10^n, were c = 0.009361... is the asymptotic density of the coreful perfect numbers (A307960).

a(11)-a(12) from Amiram Eldar, Jul 25 2025

cp's OEIS Frontend

A307821 The number of exponential abundant numbers below 10^n.

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