A366536 - OEIS

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A366536

The number of unitary divisors of the cubefree numbers (A004709).

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

(list; graph; refs; listen; history; text; internal format)

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A004709, A005117, A034444, A072048, A077610, A358040, A366440, A366537.

Similar sequences: A366534, A366538.