OFFSET
1,1
COMMENTS
From Amiram Eldar, Jun 08 2020: (Start)
Exponential abundant numbers that are odd are relatively rare: there are 235290 even exponential abundant number smaller than the first odd term, i.e., a(1) = A129575(235291).
Odd exponential abundant numbers k such that k-1 or k+1 is also exponential abundant number exist (e.g. (73#/5#)^2-1 and (73#/5#)^2 are both exponential abundant numbers, where prime(k)# = A002110(k)). Which pair is the least?
The least exponential abundant number that is coprime to 6 is (31#/3#)^2 = 1117347505588495206025. In general, the least exponential abundant number that is coprime to A002110(k) is (A007708(k+1)#/A002110(k))^2. (End)
The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 5.29...*10^(-9), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, e-Divisor.
Eric Weisstein's World of Mathematics, e-Perfect Number.
EXAMPLE
225450225 is in the sequence since it is odd and A051377(225450225) = 484323840 > 2 * 225450225.
MATHEMATICA
esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s={}; Do[If[esigma[n]>2n, AppendTo[s, n]], {n, 1, 10^10, 2}]; s
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 28 2018
STATUS
approved