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

1,1

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

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]

Prog

(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, ", "))); }

Crossrefs

Intersection of A001694 and A340109.

Subsequence of A356871.

Sequence in context: A335939 A328562 A340109 * A190106 A252572 A035902

Adjacent sequences: A364988 A364989 A364990 * A364992 A364993 A364994

Keyword

nonn

Author

Amiram Eldar, Aug 15 2023

Status

approved