%I #9 Sep 03 2022 08:49:47

%S 72,108,144,200,216,288,324,400,432,576,648,784,800,864,900,972,1000,

%T 1152,1296,1568,1600,1728,1764,1800,1936,1944,2000,2304,2592,2700,

%U 2704,2744,2916,3136,3200,3456,3528,3600,3872,3888,4000,4356,4500,4608,4900,5000,5184

%N Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).

%C For squarefree numbers k, csigma(k) = k, where csigma(k) is the sum of the coreful divisors of k (A057723). Thus, if m is a term (csigma(m) > 2*m) and k is a squarefree number coprime to k, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 2*k*m, so k*m is a coreful abundant number. Therefore, the sequence of coreful abundant numbers (A308053) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful abundant numbers can be calculated from this sequence (see comment in A308053).

%H Amiram Eldar, <a href="/A356871/b356871.txt">Table of n, a(n) for n = 1..10000</a>

%e 72 is a term since csigma(72) = 168 > 2 * 72, and 72 = 2^3 * 3^2 is powerful.

%t f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := If[AllTrue[(fct = FactorInteger[n])[[;;, 2]], #>1 &], Times @@ f @@@ fct, 0]; seq={}; Do[If[s[n] > 2*n, AppendTo[seq, n]], {n, 1, 5000}]; seq

%Y Intersection of A001694 and A308053.

%Y A339940 is a subsequence.

%Y Cf. A057723.

%Y Similar sequences: A307959, A328136.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 02 2022