A365851 - OEIS

%I #8 Nov 18 2023 13:12:08

%S 1,2,3,4,4,6,5,6,6,8,6,8,6,9,8,8,10,8,8,12,7,8,12,8,10,12,8,12,12,9,8,

%T 12,10,16,12,8,12,12,15,12,12,12,10,16,10,18,14,9,12,12,12,10,16,16,

%U 12,12,12,12,20,18,9,12,16,16,10,12,16,18,12,18,10,12

%N The number of divisors of the n-th practical number (A005153).

%C a(n) = A000005(A005153(n)).

%e A005153(1) = 1 and tau(1) = 1, so a(1) = 1.

%e A005153(2) = 2 and tau(2) = 2, so a(2) = 2.

%e A005153(3) = 4 and tau(4) = 3, so a(3) = 3.

%t f[p_, e_] := (p^(e + 1) - 1)/(p - 1); s[n_] := Module[{fct = FactorInteger[n], p, e}, p = fct[[;; , 1]]; e = fct[[;; , 2]]; If[Position[p/(1 + FoldList[Times, 1, f @@@ Most@ fct]), _?(# > 1 &)] == {}, Times @@ (e + 1), Nothing]]; s[1] = 1; Array[s, 320] (* _Amiram Eldar_, Oct 17 2023 *)

%o (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}>; a:=[]; for n in [1..400] do if ff(n) then Append(~a,#Divisors(n)); end if; end for; a;

%Y Cf. A000005, A005153.

%K nonn

%O 1,2

%A _Marius A. Burtea_, Oct 17 2023