A339939 Coreful weird numbers: numbers k that are coreful abundant (A308053) but no subset of their aliquot coreful divisors sums to k.

4900, 14700, 53900, 63700, 83300, 93100, 112700, 142100, 151900, 161700, 181300, 191100, 200900, 210700, 230300, 249900, 259700, 279300, 289100, 298900, 328300, 338100, 347900, 349448, 357700, 387100, 406700, 426300, 436100, 455700, 475300, 494900, 504700, 524300

Offset

1,1

Comments

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)

Links

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

Example

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.

Mathematica

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]

Crossrefs

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

Keyword

nonn

Author

Amiram Eldar, Dec 23 2020

Status

approved