A336253 Exponential barely deficient numbers: exponential deficient numbers whose exponential abundancy is closer to 2 than that of any smaller exponential deficient number.

1, 4, 72, 100, 144, 3528, 12100, 15876, 24336, 441000, 1334025, 2205000, 5664400, 24206400, 71267364, 151880976, 3252372552, 9346201200, 13319078472, 26828235000, 347372082000, 1851803856100, 2260121356900, 3198696480100, 5202286387272, 10330374528100, 16316106062400

Offset: 1

Author

Amiram Eldar, Jul 14 2020

Keywords

• nonn

Comments

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).

Exponential deficient numbers are numbers k with esigma(k)/k < 2. These are numbers that are neither e-perfect (A054979) nor exponential abundant (A129575).

The corresponding values of the exponential abundancy are 1, 1.5, 1.666..., 1.8..., 1.833..., ...

All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

Examples

			4 is a term since it is exponential deficient, and esigma(4)/4 = 3/2 is higher than esigma(k)/k for all the exponential deficient numbers k < 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..36

Crossrefs

Cf. A051377, A054979, A057521, A129575.

Subsequence of A001694.

Similar sequences: A302572, A228450, A262228, A307122, A336252, A336254.

Programs

Mathematica

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 0; s={}; Do[r = esigma[n]/n; If[r >= 2, Continue[]]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

Extensions

a(21)-a(27) from Amiram Eldar, May 06 2025