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
Examples
n=2: 3^2 - 1 = 8 = 2^3 so a(2)=3.
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.
Programs
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Maple
seq(padic:-ordp(3^n-1, 2), n=1..100); # Robert Israel, Dec 28 2015
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Mathematica
Table[Part[Flatten[FactorInteger[ -1+3^n]], 2], {n, 1, 70}] IntegerExponent[#,2]&/@(3^Range[110]-1) (* Harvey P. Dale, Jan 28 2017 *) -
PARI
a(n)=if(n<1,0,if(n%2==0,a(n/2)+1+(n/2)%2,1)) /* Ralf Stephan, Jan 23 2004 */
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PARI
a(n)=valuation(fibonacci(3*n),2); \\ Joerg Arndt, Oct 28 2012
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PARI
a(n)=my(t=valuation(n,2)); if(t,t+2,1) \\ Charles R Greathouse IV, Mar 14 2014
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Python
def A090740(n): return (n&-n).bit_length()+int(not n&1) # Chai Wah Wu, Jul 11 2022
Formula
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
a(n) = A337923(3*n). [Lengyel]. - R. J. Mathar, Nov 05 2008
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
Comments