A383693 - OEIS

%I #9 May 07 2025 10:49:38

%S 900,1764,4356,4500,4900,6084,6300,8820,9900,10404,11700,12348,12996,

%T 14700,15300,17100,19044,19404,20700,21780,22500,22932,26100,27900,

%U 29988,30276,30420,30492,31500,33300,33516,34596,36900,38700,40572,42300,42588,44100,47700,47916,49284,49500

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

%C First differs from its subsequence A383697 at n = 21.

%C All the terms are nonsquarefree numbers (A013929), since A322857(k) = k if k is a squarefree number (A005117).

%C 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 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 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 Amiram Eldar, <a href="/A383693/b383693.txt">Table of n, a(n) for n = 1..10000</a>

%e 900 is a term since A322857(900) = 2160 > 2*900 = 1800.

%t f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Times @@ f @@@ FactorInteger[n] > 2 n; Select[Range[50000], q]

%o (PARI) fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));

%o isok(k) = {my(f = factor(k)); prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

%Y Cf. A005117, A209061, A322857, A322858, A361255.

%Y Subsequence of A013929 and A129575.

%Y Subsequences: A383694, A383697, A383698.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, May 05 2025