cp's OEIS Frontend

A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

%I A387155 #10 Aug 20 2025 10:56:08
%S A387155 22148167706,52012,10828,24601,23660,29114,58967,118828,238600,478099,
%T A387155 957324,1916191,3834167,7669094,15335488,30667762,61337894,122679755,
%U A387155 245357929,490718137,981456651,1962956352,3925957422,7851819466,15703524589,31406984903,62813576969
%N A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.
%C A387155 n-free numbers are numbers that are not divisible by an n-th power larger than 1. E.g., A005117, A004709, and A046100 for n = 2, 3, and 4, respectively.
%C A387155 The sum of n-free divisors of a number is the sum of its divisors that are n-free numbers. E.g., A048250, A073185, and A385006 for n = 2, 3, and, respectively.
%F A387155 Let A_k(n) be the number of k-free abundant numbers that are not exceeding n. Then, a(n) = A_n(A387154(n)) - 1.
%F A387155 a(n) ~ c * 945 * 2^n, where c = A302991.
%e A387155 a(2) = 22148167706 because there are 22148167706 squarefree numbers k such that A048250(k) > 2*k (i.e., terms of A087248) that are less than the least nonsquarefree number k that has this property, A387154(2) = 401120980260.
%e A387155 a(3) = 52012 because there are 52012 cubefree numbers k such that A073185(k) > 2*k (i.e., terms of A357695) that are less than the least noncubefree number k that has this property, A387154(3) = 360360.
%t A387155 freeQ[n_, k_] := AllTrue[FactorInteger[n][[;; , 2]], # < k &];
%t A387155 sigma[n_, k_] := Times @@ ((First[#]^(Min[Last[#], k - 1] + 1) - 1)/(First[#] - 1) & /@ FactorInteger[n]);
%t A387155 a[n_] := Module[{m = 2, c = 0}, While[True, If[sigma[m, n] > 2*m, c++; If[!freeQ[m, n], Break[]]]; m++]; c-1];
%o A387155 (PARI) isfree(n, k) = if(n == 1, 1, my(e = factor(n)[,2]); for(i=1, #e, if(e[i] >= k, return(0))); 1);
%o A387155 sigmafree(n, k) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,2],k-1)+1)-1)/(f[i,1]-1));}
%o A387155 a(n) = {my(m = 2, c = 0); while(1, if(sigmafree(m, n) > 2*m, c++; if(!isfree(m, n), break)); m++); c-1;}
%Y A387155 Cf. A004709, A005117, A046099, A048250, A073185, A091194, A302991, A357695, A357700, A387154.
%K A387155 nonn,new
%O A387155 2,1
%A A387155 _Amiram Eldar_, Aug 19 2025