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A383698 Primitive exponential squarefree exponential abundant numbers: the powerful terms of A383697.

%I A383698 #8 May 07 2025 10:50:45
%S A383698 900,1764,4356,4500,4900,6084,10404,12348,12996,19044,30276,34596,
%T A383698 44100,47916,49284,60516,66564,79092,79524,88200,101124,108900,112500,
%U A383698 125316,132300,133956,152100,161604,176868,181476,191844,213444,217800,220500,224676,246924
%N A383698 Primitive exponential squarefree exponential abundant numbers: the powerful terms of A383697.
%C A383698 Subsequence of A383694 and first differs from it at n = 11.
%C A383698 The least odd term is a(1345) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.
%C A383698 For squarefree numbers k, essigma(k) = k, where essigma is the sum of exponential squarefree exponential divisors function (A361174). Thus, if m is a term (essigma(m) > 2*m) and k is a squarefree number coprime to m, then essigma(k*m) = essigma(k) * essigma(m) = k * essigma(m) > 2*k*m, so k*m is an exponential squarefree exponential abundant number. Therefore, the sequence of exponential squarefree exponential abundant numbers (A383697) can be generated from this sequence by multiplying with coprime squarefree numbers.
%H A383698 Amiram Eldar, <a href="/A383698/b383698.txt">Table of n, a(n) for n = 1..10000</a>
%t A383698 fun[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]
%o A383698 (PARI) fun(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
%o A383698 isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}
%Y A383698 Intersection of A001694 and A383697.
%Y A383698 Subsequence of A383694.
%Y A383698 Cf. A005117, A361174.
%K A383698 nonn
%O A383698 1,1
%A A383698 _Amiram Eldar_, May 06 2025