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A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

%I A340109 #15 Aug 15 2023 06:41:32
%S A340109 5400,7200,10800,14400,16200,18000,21168,21600,27000,28800,32400,
%T A340109 36000,37800,42336,43200,48600,50400,54000,56448,57600,59400,63504,
%U A340109 64800,70200,72000,75600,79200,81000,84672,86400,88200,90000,91800,93600,97200,98784,100800,102600
%N A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).
%C A340109 A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
%C A340109 Analogous to A068403 as A308053 is analogous to A005101.
%C A340109 Apparently, the least odd term in this sequence is 3^4 * 5^3 * 7^3 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29^2 = 3296233276111741840875.
%C A340109 The asymptotic density of this sequence is Sum_{n>=1} f(A364991(n)) = 0.0004006..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - _Amiram Eldar_, Aug 15 2023
%H A340109 Amiram Eldar, <a href="/A340109/b340109.txt">Table of n, a(n) for n = 1..10000</a>
%e A340109 5400 is a term since csigma(5400) = 16380 > 3 * 5400.
%t A340109 f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] > 3*# &]
%o A340109 (PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
%o A340109 is(n) = s(n) > 3*n; \\ _Amiram Eldar_, Aug 15 2023
%Y A340109 Subsequence of A308053.
%Y A340109 Cf. A007947, A057723, A364991 (primitive terms).
%Y A340109 Similar sequences: A068403, A285615, A293187, A300664, A307112, A328135.
%K A340109 nonn
%O A340109 1,1
%A A340109 _Amiram Eldar_, Dec 28 2020