A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

900, 1764, 3600, 4356, 4500, 4900, 12348, 47916, 79092, 112500, 605052, 2812500, 13366548, 29647548, 89139564, 231708348, 701538156, 1757812500, 14772192228, 32179382604, 43945312500, 71183762748, 620995547124, 990454107996, 3417547576788, 3488004374652, 10271220141996

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).

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

			The first 6 exponential abundant numbers, 900, 1764, 3600, 4356, 4500 and 4900, have decreasing values of exponential abundancy: 2.4, 2.285..., 2.2, 2.181..., 2.08, 2.057... and therefore they are in this sequence. The next exponential abundant number with a lower exponential abundancy is 12348 with eisgma(12348)/12348 = 2.040...
		

Links

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

Crossrefs

The exponential version of A071927.

Subsequence of A001694 and A328136.

Cf. A051377, A057521, A129575, A336253.

Similar sequences: A188263, A302570, A302571, A335054.

Programs

Mathematica

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 3; 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(23)-a(27) from Amiram Eldar, May 06 2025