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A129575
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Exponential abundant numbers: integers k for which A126164(k) > k, or equivalently for which A051377(k) > 2k.
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21
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900, 1764, 3600, 4356, 4500, 4900, 6084, 6300, 7056, 8100, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 25200, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 39600
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OFFSET
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1,1
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COMMENTS
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There are only 52189 exponential abundant numbers less than 50 million, which suggests that these account for approximately 0.1% of all integers.
Includes 36*m for all m coprime to 6 that are not squarefree. - Robert Israel, Feb 19 2019
The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 0.001043673..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - 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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The third integer that is exceeded by its proper exponential divisor sum is 3600. Hence a(3) = 3600.
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MAPLE
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filter:= proc(n) local L, m, i, j;
L:= ifactors(n)[2];
m:= nops(L);
mul(add(L[i][1]^j, j=numtheory:-divisors(L[i][2])), i=1..m)>2*n
end proc:
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MATHEMATICA
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ExponentialDivisors[1]={1}; ExponentialDivisors[n_]:=Module[{}, {pr, pows}=Transpose@FactorInteger[n]; divpowers=Distribute[Divisors[pows], List]; Sort[Times@@(pr^Transpose[divpowers])]]; properexponentialdivisorsum[k_]:=Plus@@ExponentialDivisors[k]-k; Select[Range[5 10^4], properexponentialdivisorsum[ # ]># &]
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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