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A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).

%I A364991 #7 Aug 15 2023 07:43:22
%S A364991 5400,7200,10800,14400,16200,18000,21168,21600,27000,28800,32400,
%T A364991 36000,42336,43200,48600,54000,56448,57600,63504,64800,72000,81000,
%U A364991 84672,86400,88200,90000,97200,98784,104544,108000,112896,115200,127008,129600,135000,144000,145800
%N A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).
%C A364991 Powerful numbers k such that csigma(k) > 3*k, where csigma(k) = A057723(k) is the sum of the coreful divisors of k.
%C A364991 If m is a term and k is a squarefree number coprime to m, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 3*k*m, so k*m is a coreful 3-abundant number. Therefore, the sequence of coreful 3-abundant numbers (A340109) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful 3-abundant numbers can be calculated from this sequence (see comment in A340109).
%H A364991 Amiram Eldar, <a href="/A364991/b364991.txt">Table of n, a(n) for n = 1..10000</a>
%t A364991 f[p_, e_] := (p^(e+1)-1)/(p-1)-1; g[1] = 1; g[n_] := If[AllTrue[(fct = FactorInteger[n])[[;; , 2]], #>1 &], Times @@ f @@@ fct, 0]; seq[kmax_] := Module[{s = {}}, Do[If[g[k] > 3*k, AppendTo[s, k]], {k, 1, kmax}]; s]; seq[500000]
%o A364991 (PARI) s(f) = prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);
%o A364991 lista(kmax) = {my(f); for(k=2, kmax, f=factor(k); if(vecmin(f[,2]) > 1 && s(f) > 3*k, print1(k, ", ")));}
%Y A364991 Intersection of A001694 and A340109.
%Y A364991 Subsequence of A356871.
%K A364991 nonn
%O A364991 1,1
%A A364991 _Amiram Eldar_, Aug 15 2023