A364769 Numbers k for which k and the arithmetic derivative k' (A003415) are practical numbers (A005153).

2, 4, 8, 12, 16, 20, 28, 32, 36, 48, 64, 72, 80, 88, 96, 100, 108, 112, 128, 144, 156, 160, 176, 180, 192, 196, 200, 208, 216, 240, 252, 256, 272, 276, 288, 300, 304, 308, 320, 324, 336, 348, 352, 380, 384, 392, 396, 400, 420, 432, 448, 456, 468, 480, 496, 500

Offset

1,1

Links

Table of n, a(n) for n=1..56.

Example

2 = A005153(2) and 2' = 1 = A005153(1), so 2 is a term.
8 = A005153(5) and 8' = 12 = A005153(6), so 8 is a term.
20 = A005153(9) and 20' = 24 = A005153(10), so 20 is a term.

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[2, 500], pracQ[#] && pracQ[d[#]] &] (* Amiram Eldar, Aug 21 2023 *)

Prog

(Magma) sk:=func<n, k|&+[Divisors(n)[i]:i in [1..k]]>; ff:=func<n|forall{k:k
in [2..#Divisors(n)]|sk(n, k-1) ge Divisors(n)[k]-1}>; f:=func<n|n le 1 select 0 else n*(&+[Factorisation(n)[i][2] /Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [n:n in [2..500]|ff(n) and ff(Floor(f(n)))];

Crossrefs

Cf. A003415, A005153.

Sequence in context: A186146 A006638 A001212 * A160742 A160736 A118030

Adjacent sequences: A364766 A364767 A364768 * A364770 A364771 A364772

Keyword

nonn

Author

Marius A. Burtea, Aug 18 2023

Status

approved