A356871 Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).

72, 108, 144, 200, 216, 288, 324, 400, 432, 576, 648, 784, 800, 864, 900, 972, 1000, 1152, 1296, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000, 4356, 4500, 4608, 4900, 5000, 5184

Offset

1,1

Comments

For squarefree numbers k, csigma(k) = k, where csigma(k) is the sum of the coreful divisors of k (A057723). Thus, if m is a term (csigma(m) > 2*m) and k is a squarefree number coprime to k, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 2*k*m, so k*m is a coreful abundant number. Therefore, the sequence of coreful abundant numbers (A308053) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful abundant numbers can be calculated from this sequence (see comment in A308053).

Links

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

Example

72 is a term since csigma(72) = 168 > 2 * 72, and 72 = 2^3 * 3^2 is powerful.

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := If[AllTrue[(fct = FactorInteger[n])[[;; , 2]], #>1 &], Times @@ f @@@ fct, 0]; seq={}; Do[If[s[n] > 2*n, AppendTo[seq, n]], {n, 1, 5000}]; seq

Crossrefs

Intersection of A001694 and A308053.

A339940 is a subsequence.

Cf. A057723.

Similar sequences: A307959, A328136.

Sequence in context: A216426 A308053 A372404 * A359280 A307758 A272191

Adjacent sequences: A356868 A356869 A356870 * A356872 A356873 A356874

Keyword

nonn

Author

Amiram Eldar, Sep 02 2022

Status

approved