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A366536
The number of unitary divisors of the cubefree numbers (A004709).
5
1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 4, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 8, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 8, 4, 4, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8
OFFSET
1,2
COMMENTS
The number of unitary divisors of the squarefree numbers (A005117) is the same as the number of divisors of the squarefree numbers (A072048), because all the divisors of a squarefree number are unitary.
LINKS
FORMULA
a(n) = A034444(A004709(n)).
MATHEMATICA
f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # < 3 &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]
PROG
(PARI) lista(max) = for(k = 1, max, my(e = factor(k)[, 2], iscubefree = 1); for(i = 1, #e, if(e[i] > 2, iscubefree = 0; break)); if(iscubefree, print1(2^(#e), ", ")));
(Python)
from sympy.ntheory.factor_ import udivisor_count
from sympy import mobius, integer_nthroot
def A366536(n):
def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
return udivisor_count(m) # Chai Wah Wu, Aug 05 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Oct 12 2023
STATUS
approved