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A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

%I A383693 #9 May 07 2025 10:49:38
%S A383693 900,1764,4356,4500,4900,6084,6300,8820,9900,10404,11700,12348,12996,
%T A383693 14700,15300,17100,19044,19404,20700,21780,22500,22932,26100,27900,
%U A383693 29988,30276,30420,30492,31500,33300,33516,34596,36900,38700,40572,42300,42588,44100,47700,47916,49284,49500
%N A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.
%C A383693 First differs from its subsequence A383697 at n = 21.
%C A383693 All the terms are nonsquarefree numbers (A013929), since A322857(k) = k if k is a squarefree number (A005117).
%C A383693 If an exponential abundant number (A129575) is exponentially squarefree (A209061), then it is in this sequence. Terms of this sequence that are not exponentially squarefree are a(21) = 22500, a(77) = 86436, a(140) = 157500, etc..
%C A383693 The least odd term is a(202273) = 225450225, and the least term that is coprime to 6 is a(1.002..*10^18) = 1117347505588495206025.
%C A383693 The asymptotic density of this sequence is Sum_{n>=1} f(A383694(n)) = 0.00089722..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).
%H A383693 Amiram Eldar, <a href="/A383693/b383693.txt">Table of n, a(n) for n = 1..10000</a>
%e A383693 900 is a term since A322857(900) = 2160 > 2*900 = 1800.
%t A383693 f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Times @@ f @@@ FactorInteger[n] > 2 n; Select[Range[50000], q]
%o A383693 (PARI) fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
%o A383693 isok(k) = {my(f = factor(k)); prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}
%Y A383693 Cf. A005117, A209061, A322857, A322858, A361255.
%Y A383693 Subsequence of A013929 and A129575.
%Y A383693 Subsequences: A383694, A383697, A383698.
%K A383693 nonn,easy
%O A383693 1,1
%A A383693 _Amiram Eldar_, May 05 2025