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A358039
a(n) is the Euler totient function phi applied to the n-th cubefree number.
4
1, 1, 2, 2, 4, 2, 6, 6, 4, 10, 4, 12, 6, 8, 16, 6, 18, 8, 12, 10, 22, 20, 12, 12, 28, 8, 30, 20, 16, 24, 12, 36, 18, 24, 40, 12, 42, 20, 24, 22, 46, 42, 20, 32, 24, 52, 40, 36, 28, 58, 16, 60, 30, 36, 48, 20, 66, 32, 44, 24, 70, 72, 36, 40, 36, 60, 24, 78, 40
OFFSET
1,3
COMMENTS
The analogous sequence with squarefree numbers is A049200.
LINKS
Zhu Weiyi, On the cube free number sequences, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.
FORMULA
a(n) = A000010(A004709(n)).
Sum_{k=1..n} a(k) = (c/(2*zeta(3)))*n^2 + O(n^(3/2+eps)), where c = Product_{p prime} (1 - (p+1)/(p^3+p^2+1)) = 0.62583324412633345811... (Weiyi, 2004).
From Amiram Eldar, Oct 09 2023: (Start)
Sum_{n>=1} 1/(A004709(n)*a(n)) = Product_{p prime} (1 + (p^2+1)/((p-1)*p^3)) = 2.14437852780769816048... .
Sum_{n>=1} 1/a(n)^2 = Product_{p prime} (1 + (p^2+1)/((p-1)^2*p^2)) = 3.26032746607943673536... . (End)
MATHEMATICA
EulerPhi[Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]]
PROG
(Python)
from sympy import mobius, integer_nthroot, totient
def A358039(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 totient(m) # Chai Wah Wu, Aug 06 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 29 2022
STATUS
approved