cp's OEIS Frontend

A383694 Primitive exponential unitary abundant numbers: the powerful terms of A383693.

%I A383694 #8 May 07 2025 10:49:58
%S A383694 900,1764,4356,4500,4900,6084,10404,12348,12996,19044,22500,30276,
%T A383694 34596,44100,47916,49284,60516,66564,79092,79524,86436,88200,101124,
%U A383694 108900,112500,125316,132300,133956,152100,161604,176400,176868,181476,191844,213444,217800,220500
%N A383694 Primitive exponential unitary abundant numbers: the powerful terms of A383693.
%C A383694 First differs from its subsequence A383698 at n = 11.
%C A383694 For squarefree numbers k, eusigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857). Thus, if m is a term (eusigma(m) > 2*m) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) > 2*k*m, so k*m is an exponential unitary abundant number. Therefore, the sequence of exponential unitary abundant numbers (A383693) can be generated from this sequence by multiplying with coprime squarefree numbers.
%C A383694 The least odd term is a(1455) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.
%H A383694 Amiram Eldar, <a href="/A383694/b383694.txt">Table of n, a(n) for n = 1..10000</a>
%e A383694 900 is a term since eusigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is a powerful number.
%e A383694 6300 is exponential unitary abundant, since eusigma(6300) = 15120 > 2 * 6300, but it is not a powerful number: 6300 = 2^2 * 3^2 * 5^2 * 7. Thus it is not in this sequence. It can be generated as a term of A383693 from a(1) = 900 by 7 * 900 = 6300, since 7 is squarefree and gcd(7, 900) = 1.
%t A383694 fun[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]
%o A383694 (PARI) fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
%o A383694 isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}
%Y A383694 Intersection of A001694 and A383693.
%Y A383694 A383698 is a subsequence.
%Y A383694 Cf. A005117, A322857, A328136.
%K A383694 nonn
%O A383694 1,1
%A A383694 _Amiram Eldar_, May 05 2025