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A244413
Exponent of highest power of 8 dividing n.
10
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
OFFSET
1,64
COMMENTS
This is the member g = 8 in the g-family of sequences, g integer >= 2, call it phi(g,n), n >= 1. In the Mahler reference, Lemma 2, pp. 6-7, this exponent is called f = -phi if g divides r = n (s = 1 there), and f = 0 if g does not divide r = n (s = 1 there).
REFERENCES
Kurt Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.
LINKS
FORMULA
n = 8^a(n)*m with a(n) nonnegative integer such that 8 does not divide m, for n >= 1.
O.g.f.: Sum_{k>=1} x^(8^k)/(1-x^(8^k)).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/7. - Amiram Eldar, Jan 17 2022
a(n) = floor(A007814(n)/3). - Alan Michael Gómez Calderón, Jul 25 2024
MATHEMATICA
Table[IntegerExponent[n, 8], {n, 1, 100}] (* Amiram Eldar, Sep 14 2020 *)
PROG
(PARI) A244413(n) = valuation(n, 8); \\ Antti Karttunen, Oct 07 2017
(Python)
def A244413(n): return (~n&n-1).bit_length()//3 # Chai Wah Wu, Jul 09 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 27 2014
STATUS
approved