A090740 Exponent of 2 in 3^n - 1.

1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 8, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 5, 1

Offset

1,2

Comments

Also the 2-adic order of Fibonacci(3n) [Lengyel]. - R. J. Mathar, Nov 05 2008

Links

Robert Israel, Table of n, a(n) for n = 1..10000

T. Lengyel, The order of the Fibonacci and Lucas numbers, Fib. Quart. 33 (1995), 234-239.

Diego Marques and Pavel Trojovský, The p-adic order of some fibonomial coefficients, J. Int. Seq. 18 (2015), Article 15.3.1, proposition 7.

Formula

a(n) = A007814(n) + A059841(n) + 1.

Multiplicative with a(p^e) = e+2 if p = 2; 1 if p > 2. G.f.: A(x) = 1/(1-x^2) + Sum_{k>=0} x^(2^k)/(1-x^(2^k)). - Vladeta Jovovic, Jan 19 2004

G.f.: Sum_{k>=0} t*(1+2*t+t^2+t^3)/(1-t^4) with t=x^2^k. Recurrence: a(2n) = a(n) + 1 + [n odd], a(2n+1) = 1. - Ralf Stephan, Jan 23 2004

G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x) + x^2/(1-x^4). - Robert Israel, Dec 28 2015

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2. - Amiram Eldar, Nov 28 2022

Dirichlet g.f.: zeta(s)*(2^s+1-1/2^s)/(2^s-1). - Amiram Eldar, Jan 04 2023

Example

n=2: 3^2 - 1 = 8 = 2^3 so a(2)=3.

Maple

seq(padic:-ordp(3^n-1, 2), n=1..100); # Robert Israel, Dec 28 2015

Mathematica

Table[Part[Flatten[FactorInteger[ -1+3^n]], 2], {n, 1, 70}]
IntegerExponent[#, 2]&/@(3^Range[110]-1) (* Harvey P. Dale, Jan 28 2017 *)

Prog

(PARI) a(n)=if(n<1, 0, if(n%2==0, a(n/2)+1+(n/2)%2, 1)) /* Ralf Stephan, Jan 23 2004 */
(PARI) a(n)=valuation(fibonacci(3*n), 2); \\ Joerg Arndt, Oct 28 2012
(PARI) a(n)=my(t=valuation(n, 2)); if(t, t+2, 1) \\ Charles R Greathouse IV, Mar 14 2014
(Python)
def A090740(n): return (n&-n).bit_length()+int(not n&1) # Chai Wah Wu, Jul 11 2022

Crossrefs

Cf. A069895, A091512, A088660, A090739, A001511.

Cf. A007814, A014445, A059841.

Sequence in context: A004592 A116992 A233267 * A094603 A165595 A213181

Adjacent sequences: A090737 A090738 A090739 * A090741 A090742 A090743

Keyword

nonn,mult,easy

Author

Labos Elemer and Ralf Stephan, Jan 19 2004

Status

approved