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A356871
Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).
5
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
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 A376720
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 02 2022
STATUS
approved