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A129575 Exponential abundant numbers: integers k for which A126164(k) > k, or equivalently for which A051377(k) > 2k.

%I A129575 #21 May 06 2025 09:33:18
%S A129575 900,1764,3600,4356,4500,4900,6084,6300,7056,8100,8820,9900,10404,
%T A129575 11700,12348,12996,14700,15300,17100,19044,19404,20700,21780,22500,
%U A129575 22932,25200,26100,27900,29988,30276,30420,30492,31500,33300,33516,34596,35280,36900,38700,39600
%N A129575 Exponential abundant numbers: integers k for which A126164(k) > k, or equivalently for which A051377(k) > 2k.
%C A129575 There are only 52189 exponential abundant numbers less than 50 million, which suggests that these account for approximately 0.1% of all integers.
%C A129575 Includes 36*m for all m coprime to 6 that are not squarefree. - _Robert Israel_, Feb 19 2019
%C A129575 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
%H A129575 Robert Israel, <a href="/A129575/b129575.txt">Table of n, a(n) for n = 1..10000</a>
%H A129575 Peter Hagis, Jr., <a href="https://doi.org/10.1155/S0161171288000407">Some Results Concerning Exponential Divisors</a>, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
%H A129575 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>.
%e A129575 The third integer that is exceeded by its proper exponential divisor sum is 3600. Hence a(3) = 3600.
%p A129575 filter:= proc(n)  local L,m,i,j;
%p A129575   L:= ifactors(n)[2];
%p A129575   m:= nops(L);
%p A129575   mul(add(L[i][1]^j, j=numtheory:-divisors(L[i][2])),i=1..m)>2*n
%p A129575 end proc:
%p A129575 select(filter, [$1..10^5]); # _Robert Israel_, Feb 19 2019
%t A129575 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[ # ]># &]
%t A129575 f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[40000], esigma[#] > 2*# &] (* _Amiram Eldar_, May 06 2025 *)
%o A129575 (PARI) isok(k) = {my(f = factor(k)); prod(i = 1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d)) > 2*k;} \\ _Amiram Eldar_, May 06 2025
%Y A129575 Cf. A126164, A051377, A049419, A054979, A054980, A328136.
%K A129575 easy,nonn
%O A129575 1,1
%A A129575 _Ant King_, Apr 28 2007