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A328136 Primitive exponential abundant numbers: the powerful terms of A129575.

%I A328136 #10 Oct 05 2019 04:25:04
%S A328136 900,1764,3600,4356,4500,4900,6084,7056,8100,10404,12348,12996,19044,
%T A328136 22500,30276,34596,44100,47916,49284,60516,66564,79092,79524,86436,
%U A328136 88200,101124,108900,112500,125316,132300,133956,152100,161604,176400,176868,181476,191844
%N A328136 Primitive exponential abundant numbers: the powerful terms of A129575.
%C A328136 For squarefree numbers k, esigma(k) = k, where esigma is the sum of exponential divisors function (A051377). Thus, if m is a term (esigma(m) > 2m) and k is a squarefree number coprime to m, then esigma(k*m) = esigma(k) * esigma(m) = k * esigma(m) > 2*k*m, so k*m is an exponential abundant number. Therefore the sequence of exponential abundant numbers (A129575) can be generated from this sequence by multiplying with coprime squarefree numbers.
%H A328136 Amiram Eldar, <a href="/A328136/b328136.txt">Table of n, a(n) for n = 1..10000</a>
%H A328136 Peter Hagis, <a href="http://dx.doi.org/10.1155/S0161171288000407">Some results concerning exponential divisors</a>, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2 (1988), pp. 343-349.
%H A328136 E. G. Straus and M. V. Subbarao, <a href="https://www.utdallas.edu/~m.vidyasagar/Straus-Subbarao-74.pdf">On exponential divisors</a>, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465-471.
%e A328136 900 is a term since esigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is powerful.
%e A328136 6300 is exponential abundant, since esigma(6300) = 15120 > 2 * 6300, but it is not powerful, 6300 = 2^2 * 3^2 * 5^2 * 7, thus it is not in this sequence. It can be generated as a term of A129575 from 900 by 7 * 900 = 6300, since gcd(7, 900) = 1.
%t A328136 fun[p_, e_] := DivisorSum[e, p^# &];aQ[n_] := Min[(f = FactorInteger[n])[[;;,2]]] > 1 && Times @@ fun @@@ f > 2n; Select[Range[200000], aQ]
%Y A328136 Intersection of A001694 and A129575.
%Y A328136 Cf. A051377, A054979, A054980, A126164.
%K A328136 nonn
%O A328136 1,1
%A A328136 _Amiram Eldar_, Oct 04 2019