login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A264981 Highest power of 9 dividing n. 2
1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 81, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 9. - Tom Edgar, Feb 02 2016
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) = 9^valuation(n,9). - Tom Edgar, Feb 02 2016
G.f.: x/(1 - x) + 8 * Sum_{k>=1} 9^(k-1)*x^(9^k)/(1 - x^(9^k)). - Ilya Gutkovskiy, Jul 10 2019
From Amiram Eldar, Dec 31 2022: (Start)
Multiplicative with a(3^e) = 3^(2*floor(e/2)), and a(p^e) = 1 if p != 3.
Dirichlet g.f.: zeta(s)*(9^s-1)/(9^s-9).
Sum_{k=1..n} a(k) ~ (4/(9*log(3)))*n*log(n) + (5/9 + 4*(gamma-1)/(9*log(3)))*n, where gamma is Euler's constant (A001620). (End)
EXAMPLE
Since 18 = 9 * 2, a(18) = 9. Likewise, since 9 does not divide 17, a(17) = 1. - Tom Edgar, Feb 02 2016
MATHEMATICA
Table[9^Length@ TakeWhile[Reverse@ IntegerDigits[n, 9], # == 0 &], {n, 99}] (* Michael De Vlieger, Dec 09 2015 *)
9^Table[IntegerExponent[n, 9], {n, 150}] (* Vincenzo Librandi, Feb 03 2016 *)
PROG
(Scheme)
(define (A264981 n) (let loop ((k 9)) (if (not (zero? (modulo n k))) (/ k 9) (loop (* 9 k)))))
(PARI) a(n) = 9^valuation(n, 9); \\ Michel Marcus, Dec 08 2015
(Sage) [9^valuation(i, 9) for i in [1..100]] # Tom Edgar, Feb 02 2016
CROSSREFS
Similar sequences for other bases: A006519 (2), A038500 (3), A234957 (4), A060904 (5), A234959 (6).
Sequence in context: A010534 A078297 A189788 * A370240 A113061 A366904
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Dec 07 2015
EXTENSIONS
Keyword:mult added by Andrew Howroyd, Jul 20 2018
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)