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A321147
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Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.
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11
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225450225, 385533225, 481583025, 538472025, 672624225, 705699225, 985646025, 1121915025, 1150227225, 1281998025, 1566972225, 1685513025, 1790559225, 1826280225, 2105433225, 2242496025, 2466612225, 2550755025, 2679615225, 2930852925, 2946861225, 3132081225
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Eric Weisstein's World of Mathematics, e-Divisor.
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EXAMPLE
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225450225 is in the sequence since it is odd and A051377(225450225) = 484323840 > 2 * 225450225.
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MATHEMATICA
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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
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CROSSREFS
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The exponential version of A005231.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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