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A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

%I A387153 #13 Aug 20 2025 10:56:20
%S A387153 30030,39270,43890,46410,51870,53130,62790,66990,67830,71610,79170,
%T A387153 82110,84630,85470,91770,94710,99330,101010,103530,108570,111930,
%U A387153 117390,122430,128310,136290,140910,144690,154770,161070,164010,166530,168630,182490,191730,205590
%N A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.
%C A387153 First differs from A067885 at n = 11: A067885(11) = 72930 is not a term of this sequence. a(59) = 510510 is the least term of this sequence that is not in A067885.
%C A387153 Subsequence of A285615 and first differs from it at n = 51: A285615(51) = 390390 is not a term of this sequence.
%C A387153 This sequence is not the same as the sequence of numbers k such that A048250(k) > 3*k which includes all the terms of this sequence but also nonsquarefree numbers, the least of them is 2*A002110(52) = A088860(52) = 2.1248...*10^96.
%C A387153 The least odd term is A002110(17)/2 = 961380175077106319535, the least term that is not divisible by 3 is a(5607800) = 66853496710, and the least term that is coprime to 6 is A002110(52)/6 = 1.7706...*10^95.
%C A387153 If k is a term and m is a squarefree number coprime to k, then k*m is also a term.
%C A387153 The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 17, 95, 795, 8162, 86331, 854164, 8372782, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00008... .
%H A387153 Amiram Eldar, <a href="/A387153/b387153.txt">Table of n, a(n) for n = 1..10000</a>
%F A387153 A001221(a(n)) >= 6.
%e A387153 30030 = 2 * 3 * 5 * 7 * 11 * 13 is a term since it is squarefree, and sigma(30030) = 96768 > 3*30030 = 90090.
%t A387153 q[k_] := Module[{f = FactorInteger[k]}, Max[f[[;;, 2]]] == 1 && Times @@ (1 + f[[;; , 1]]) > 3*k]; Select[Range[2*10^5], q]
%o A387153 (PARI) isok(k) = {my(f = factor(k)); issquarefree(f) && vecprod(apply(x -> x+1, f[, 1])) > 3*k;}
%Y A387153 Intersection of A005117 and A068403.
%Y A387153 Subsequence of A087248 and A285615.
%Y A387153 Cf. A000203, A001221, A002110, A013929, A048250, A088860.
%K A387153 nonn,new
%O A387153 1,1
%A A387153 _Amiram Eldar_, Aug 19 2025