%I #15 Sep 02 2022 06:10:06

%S 72,108,144,200,216,288,324,360,400,432,504,540,576,600,648,720,756,

%T 784,792,800,864,900,936,972,1000,1008,1080,1152,1188,1200,1224,1296,

%U 1368,1400,1404,1440,1512,1568,1584,1600,1620,1656,1728,1764,1800,1836,1872,1936

%N Coreful abundant numbers: numbers k such that csigma(k) > 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

%C Analogous to A005101 as A307958 is analogous to A000396.

%C The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 0.0262215..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - _Amiram Eldar_, Sep 02 2022

%H Amiram Eldar, <a href="/A308053/b308053.txt">Table of n, a(n) for n = 1..10000</a>

%e 72 is in the sequence since its coreful divisors are 6, 12, 18, 24, 36, 72, whose sum is 168 > 2 * 72.

%t f[p_, e_] := (p^(e+1)-1)/(p-1)-1; a[1]=1; a[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[If[a[n] > 2n, AppendTo[s, n]], {n, 1, 2000}]; s

%o (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947

%o s(n) = rad(n)*sigma(n/rad(n)); \\ A057723

%o isok(k) = s(k) > 2*k; \\ _Michel Marcus_, May 11 2019

%o (PARI) isok(k) = {my(f=factor(k)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1)-1) > 2*k}; \\ _Amiram Eldar_, Sep 02 2022

%Y Cf. A005101, A057723, A307958.

%Y A339940 and A356871 are subsequences.

%Y Subsequence of A129575.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 10 2019