A268355 Highest power of 8 dividing n.

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

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

Cf. A001620, A006519, A038500, A234957, A244413, A234959.

Sequence in context: A367513 A368540 A103760 * A008834 A368170 A366906

Adjacent sequences: A268352 A268353 A268354 * A268356 A268357 A268358

Keyword

nonn,easy,mult

Author

Tom Edgar, Feb 02 2016

Extensions

Keyword:mult added by Andrew Howroyd, Jul 20 2018

Status

approved