A364767 The number of divisors of n that are practical numbers (A005153).

1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 5, 1, 2, 1, 5, 1, 4, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 1, 4, 1, 6, 1, 2, 1, 7, 1, 2, 1, 6, 1, 4, 1, 3, 1, 2, 1, 9, 1, 2, 1, 3, 1, 5, 1, 6, 1, 2, 1, 8, 1, 2, 1, 7, 1, 4, 1, 3, 1, 2, 1, 10, 1, 2, 1, 3, 1, 4, 1, 8, 1, 2, 1, 8, 1, 2, 1, 5, 1, 6, 1, 3, 1, 2, 1, 11, 1, 2, 1, 5, 1, 3, 1, 5, 1

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(p) = 1 for p prime, p > 2.

a(2*p) = 2 for p prime, p > 3.

a(2*3^k) = k + 2, k >= 1;

a(2*p^k) = 2, k >= 1, p prime, p >= 5.

a(2^n) = n + 1.

a(n) = Sum_{d|n} A322860(d). - Antti Karttunen, Sep 11 2023

Example

n = 1 has only one divisor 1 = A005153(1).
n = 2 has two divisors 1 = A005153(1), 2 = A005153(2).
n = 4 has three divisors 1 = A005153(1), 2 = A005153(2), 4 = A005153(3).

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 &)]) == {}; a[n_] := DivisorSum[n, 1 &, pracQ[#] &]; Array[a, 100] (* Amiram Eldar, Aug 21 2023 *)

Prog

(Magma) sk:=func<n, k|&+[Divisors(n)[i]:i in [1..k]]>; f:=func<n|forall{k: k in [2..#Divisors(n)]|sk(n, k-1) ge Divisors(n)[k]-1}>; [#[d:d in Divisors(n)|f(d)]:n in [1..100]];
(PARI) \\ using is_A005153 from A005153;
a(n) = sumdiv(n, d, is_A005153(d)); \\ Michel Marcus, Sep 11 2023

Crossrefs

Cf. A000005, A005153, A364768.

Inverse Möbius transform of A322860.

Sequence in context: A342241 A322584 A356224 * A326154 A306248 A361788

Adjacent sequences: A364764 A364765 A364766 * A364768 A364769 A364770

Keyword

nonn

Author

Marius A. Burtea, Aug 18 2023

Extensions

Data section extended up to a(105) by Antti Karttunen, Jun 02 2024

Status

approved