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
Antti Karttunen, Table of n, a(n) for n = 1..65537
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