A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 42336, 43200, 48600, 54000, 56448, 57600, 63504, 64800, 72000, 81000, 84672, 86400, 88200, 90000, 97200, 98784, 104544, 108000, 112896, 115200, 127008, 129600, 135000, 144000, 145800

Offset: 1

Author

Amiram Eldar, Aug 15 2023

Keywords

• nonn

Comments

Powerful numbers k such that csigma(k) > 3*k, where csigma(k) = A057723(k) is the sum of the coreful divisors of k.

If m is a term and k is a squarefree number coprime to m, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 3*k*m, so k*m is a coreful 3-abundant number. Therefore, the sequence of coreful 3-abundant numbers (A340109) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful 3-abundant numbers can be calculated from this sequence (see comment in A340109).

Links

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

Crossrefs

Intersection of A001694 and A340109.

Subsequence of A356871.

Programs

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; g[1] = 1; g[n_] := If[AllTrue[(fct = FactorInteger[n])[[;; , 2]], #>1 &], Times @@ f @@@ fct, 0]; seq[kmax_] := Module[{s = {}}, Do[If[g[k] > 3*k, AppendTo[s, k]], {k, 1, kmax}]; s]; seq[500000]

PARI

s(f) = prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);
lista(kmax) = {my(f); for(k=2, kmax, f=factor(k); if(vecmin(f[,2]) > 1 && s(f) > 3*k, print1(k, ", ")));}