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A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

%I A328135 #22 May 06 2025 09:34:22
%S A328135 901800900,1542132900,1926332100,2153888100,2690496900,2822796900,
%T A328135 3942584100,4487660100,4600908900,5127992100,6267888900,6742052100,
%U A328135 7162236900,7305120900,8421732900,8969984100,9866448900,10203020100,10718460900,11723411700,11787444900,12528324900
%N A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).
%C A328135 Aiello et al. found bounds on e-multiperfect numbers, i.e., numbers m such that esigma(m) = k * m for k > 2: 2 * 10^7 for k = 3, and 10^85, 10^320, and 10^1210 for k = 4, 5, and 6. The data of this sequence raise the bound for exponential 3-perfect numbers to 3 * 10^10.
%C A328135 The least odd term is (59#/2)^2 = 924251841031287598942273821762233522616225. The least term which is coprime to 6 is (239#/6)^2 = 3.135... * 10^190.
%C A328135 The least exponential 4-abundant number (esigma(m) >= 4m) is (31#)^2 = 40224510201185827416900. In general, the least exponential k-abundant number (esigma(m) >= k*m), for k > 2, is (A002110(A072986(k)))^2.
%C A328135 The asymptotic density of this sequence is Sum_{n>=1} f(A383699(n)) = 1.325...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - _Amiram Eldar_, May 06 2025
%H A328135 Amiram Eldar, <a href="/A328135/b328135.txt">Table of n, a(n) for n = 1..10000</a>
%H A328135 W. Aiello, G. E. Hardy, and M. V. Subbarao, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-1/aiello.pdf">On the existence of e-multiperfect numbers</a>, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 65-71.
%t A328135 f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^10], esigma[#] >= 3 # &]
%Y A328135 Subsequence of A129575.
%Y A328135 A383699 is a subsequence.
%Y A328135 Cf. A002110, A051377, A072986, A321147.
%Y A328135 Cf. A023197, A307112, A285615 (unitary), A293187 (bi-unitary), A300664 (infinitary).
%K A328135 nonn
%O A328135 1,1
%A A328135 _Amiram Eldar_, Oct 04 2019