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A268355
Highest power of 8 dividing n.
1
1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,8
COMMENTS
The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 8.
LINKS
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
FORMULA
a(n) = 8^valuation(n,8).
a(n) = 8^A244413(n).
G.f.: Sum_{m>=0} 8^m * Sum_{j=1..7} x^(j*8^m)/(1-x^(8^(m+1))). - Robert Israel, Feb 03 2016
From Amiram Eldar, Dec 31 2022: (Start)
Multiplicative with a(2^e) = 2^(3*floor(e/3)), and a(p^e) = 1 if p >= 3.
Dirichlet g.f.: zeta(s)*(8^s-1)/(8^s-8).
Sum_{k=1..n} a(k) ~ (7/(24*log(2)))*n*log(n) + (9/16 + 7*(gamma-1)/(24*log(2)))*n, where gamma is Euler's constant (A001620). (End)
EXAMPLE
Since 16 = 8 * 2, a(16) = 8. Likewise, since 8 does not divide 15, a(15) = 1.
MAPLE
seq(8^floor(padic:-ordp(n, 2)/3), n=1..100); # Robert Israel, Feb 03 2016
MATHEMATICA
8^Table[IntegerExponent[n, 8], {n, 150}] (* Vincenzo Librandi, Feb 03 2016 *)
PROG
(Sage) [8^valuation(i, 8) for i in [1..100]]
(PARI) a(n) = 8^valuation(n, 8); \\ Michel Marcus, Feb 05 2016
(Python)
def A268355(n): return (m:=(~n&n-1))+1>>(m.bit_length()%3) # Chai Wah Wu, Jul 09 2022
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Tom Edgar, Feb 02 2016
EXTENSIONS
Keyword:mult added by Andrew Howroyd, Jul 20 2018
STATUS
approved