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A364767
The number of divisors of n that are practical numbers (A005153).
2
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
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
Inverse Möbius transform of A322860.
Sequence in context: A342241 A322584 A356224 * A326154 A306248 A361788
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