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A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.

%I A321147 #20 Feb 16 2025 08:33:57
%S A321147 225450225,385533225,481583025,538472025,672624225,705699225,
%T A321147 985646025,1121915025,1150227225,1281998025,1566972225,1685513025,
%U A321147 1790559225,1826280225,2105433225,2242496025,2466612225,2550755025,2679615225,2930852925,2946861225,3132081225
%N A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.
%C A321147 From _Amiram Eldar_, Jun 08 2020: (Start)
%C A321147 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).
%C A321147 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?
%C A321147 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)
%C A321147 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
%H A321147 Amiram Eldar, <a href="/A321147/b321147.txt">Table of n, a(n) for n = 1..10000</a>
%H A321147 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>.
%H A321147 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-PerfectNumber.html">e-Perfect Number</a>.
%e A321147 225450225 is in the sequence since it is odd and A051377(225450225) = 484323840 > 2 * 225450225.
%t A321147 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
%Y A321147 The exponential version of A005231.
%Y A321147 The odd subsequence of A129575.
%Y A321147 Cf. A002110, A007708, A051377, A126164, A129485, A293186, A328136.
%K A321147 nonn
%O A321147 1,1
%A A321147 _Amiram Eldar_, Oct 28 2018