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A358040
a(n) is the number of divisors of the n-th cubefree number.
7
1, 2, 2, 3, 2, 4, 2, 3, 4, 2, 6, 2, 4, 4, 2, 6, 2, 6, 4, 4, 2, 3, 4, 6, 2, 8, 2, 4, 4, 4, 9, 2, 4, 4, 2, 8, 2, 6, 6, 4, 2, 3, 6, 4, 6, 2, 4, 4, 4, 2, 12, 2, 4, 6, 4, 8, 2, 6, 4, 8, 2, 2, 4, 6, 6, 4, 8, 2, 4, 2, 12, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 2, 6, 6, 9, 2
OFFSET
1,2
COMMENTS
The analogous sequence with squarefree numbers is A072048.
LINKS
Zhu Weiyi, On the cube free number sequences, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.
FORMULA
a(n) = A000005(A004709(n)).
Sum_{k=1..n} a(k) = (36*c_1/Pi^4) * n * (log(n) + (2*gamma - 1) - 24*zeta'(2)/Pi^2 - 4*c_2) + O(n^(1/2 + eps)), where c_1 = Product_{p prime} ((p^2+2*p+3)/(p+1)^2) = 1.58095136661854869148023... and c_2 = Sum_{p prime} p*log(p)/((p+1)*(p^2+2*p+3)) = 0.229224... (Weiyi, 2004).
MATHEMATICA
DivisorSigma[0, Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]]
PROG
(Python)
from sympy import mobius, integer_nthroot, divisor_count
def A358040(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 divisor_count(m) # Chai Wah Wu, Aug 06 2024
CROSSREFS
Cf. A000005, A001620 (gamma), A004709, A072048, A073002 (-zeta'(2)), A147533 (2*gamma-1), A358039.
Sequence in context: A369890 A373982 A334762 * A305461 A043261 A157986
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 29 2022
STATUS
approved