1,1

First differs from A321146 at n = 24.

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

From Amiram Eldar, Dec 01 2025: (Start)

A number k is a term if and only if k divided by its squarefree kernel is a weird number, i.e., A003557(k) is in A006037.

If k is a term and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms in this sequence (A391142) are the powerful (A001694) terms. All the terms are of the form k*m where k is primitive and m is a squarefree number coprime to k.

The asymptotic density of this sequence is Sum_{n>=1} f(A391142(n)) = Sum_{n>=1} 1/A181797(A006037(n)) = 6.1509776...*10^(-5), where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

4900 is a term since the sum of its aliquot coreful divisors, {70, 140, 350, 490, 700, 980, 2450}, is 5180 > 4900, and no subset of these divisors sums to 4900.

corDiv[n_] := Module[{rad = Times @@ FactorInteger [n][[;; , 1]]}, rad * Divisors[n/rad]]; corWeirdQ[n_] := Module[{d = Most@corDiv[n], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^5], corWeirdQ]

Subsequence of A308053.

A391142 is a subsequence.

Cf. A001694, A003557, A005117, A006037, A007947, A057723, A181797.

Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948.

Sequence in context: A251325 A186478 A220059 * A321146 A391142 A374500

Adjacent sequences: A339936 A339937 A339938 * A339940 A339941 A339942

Amiram Eldar, Dec 23 2020

approved