login
A365851
The number of divisors of the n-th practical number (A005153).
0
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, 12, 10, 16, 12, 8, 12, 12, 15, 12, 12, 12, 10, 16, 10, 18, 14, 9, 12, 12, 12, 10, 16, 16, 12, 12, 12, 12, 20, 18, 9, 12, 16, 16, 10, 12, 16, 18, 12, 18, 10, 12
OFFSET
1,2
COMMENTS
a(n) = A000005(A005153(n)).
EXAMPLE
A005153(1) = 1 and tau(1) = 1, so a(1) = 1.
A005153(2) = 2 and tau(2) = 2, so a(2) = 2.
A005153(3) = 4 and tau(4) = 3, so a(3) = 3.
MATHEMATICA
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 *)
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}>; a:=[]; for n in [1..400] do if ff(n) then Append(~a, #Divisors(n)); end if; end for; a;
CROSSREFS
Sequence in context: A178031 A008328 A298933 * A091860 A333995 A301764
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Oct 17 2023
STATUS
approved