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.

22148167706, 52012, 10828, 24601, 23660, 29114, 58967, 118828, 238600, 478099, 957324, 1916191, 3834167, 7669094, 15335488, 30667762, 61337894, 122679755, 245357929, 490718137, 981456651, 1962956352, 3925957422, 7851819466, 15703524589, 31406984903, 62813576969

Offset

2,1

Comments

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.

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.

Links

Table of n, a(n) for n=2..28.

Formula

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.

a(n) ~ c * 945 * 2^n, where c = A302991.

Example

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.
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.

Mathematica

freeQ[n_, k_] := AllTrue[FactorInteger[n][[;; , 2]], # < k &];
sigma[n_, k_] := Times @@ ((First[#]^(Min[Last[#], k - 1] + 1) - 1)/(First[#] - 1) & /@ FactorInteger[n]);
a[n_] := Module[{m = 2, c = 0}, While[True, If[sigma[m, n] > 2*m, c++; If[!freeQ[m, n], Break[]]]; m++]; c-1];

Prog

(PARI) isfree(n, k) = if(n == 1, 1, my(e = factor(n)[, 2]); for(i=1, #e, if(e[i] >= k, return(0))); 1);
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)); }
a(n) = {my(m = 2, c = 0); while(1, if(sigmafree(m, n) > 2*m, c++; if(!isfree(m, n), break)); m++); c-1; }

Crossrefs

Cf. A004709, A005117, A046099, A048250, A073185, A091194, A302991, A357695, A357700, A387154.

Sequence in context: A101816 A186910 A129250 * A216398 A121841 A306503

Adjacent sequences: A387152 A387153 A387154 * A387156 A387157 A387158

Keyword

nonn

Author

Amiram Eldar, Aug 19 2025

Status

approved