A337923 -id:A337923 - cp's OEIS frontend

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

Labos Elemer and Ralf Stephan, Jan 19 2004

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

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

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.

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

Cf. A007814, A014445, A059841.

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

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

a(n)=if(n<1,0,if(n%2==0,a(n/2)+1+(n/2)%2,1)) /* Ralf Stephan, Jan 23 2004 */

a(n)=valuation(fibonacci(3*n),2); \\ Joerg Arndt, Oct 28 2012

a(n)=my(t=valuation(n,2)); if(t,t+2,1) \\ Charles R Greathouse IV, Mar 14 2014

def A090740(n): return (n&-n).bit_length()+int(not n&1) # Chai Wah Wu, Jul 11 2022

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
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

A385458 Triangle read by rows: T(n,k) = exponent of the highest power of 2 dividing each Fibonomial coefficient fibonomial(n, k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 2, 3, 3, 0, 0, 0, 3, 2, 2, 3, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 1, 1, 0, 3, 3, 0, 1, 1, 0, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 3, 4, 4, 1, 4, 4, 3, 4, 4, 0

Offset: 0

David Radcliffe, Jun 29 2025

			Triangle begins:
   n\k  0  1  2  3  4  5  6  7  8  9 10 11 12
   0:   0
   1:   0  0
   2:   0  0  0
   3:   0  1  1  0
   4:   0  0  1  0  0
   5:   0  0  0  0  0  0
   6:   0  3  3  2  3  3  0
   7:   0  0  3  2  2  3  0  0
   8:   0  0  0  2  2  2  0  0  0
   9:   0  1  1  0  3  3  0  1  1  0
  10:   0  0  1  0  0  3  0  0  1  0  0
  11:   0  0  0  0  0  0  0  0  0  0  0  0
  12:   0  4  4  3  4  4  1  4  4  3  4  4  0

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.
Romeo Meštrović, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
David Radcliffe, Matrix plot of the first 768 rows
Diana L. Wells, The Fibonacci and Lucas triangles modulo 2, Fibonacci Quart. 32, no. 2 (1994), 111-123. (Theorem 2)

Cf. A000120, A010048, A007814, A065040, A337923, A385456, A385457, A385608.

function T_row(n)
    function T(n, k)
        c(a, b) = 2 * a + b ÷ 6 - count_ones(a)
        (nd, nm) = divrem(n, 3)
        (kd, km) = divrem(k, 3)
        !(nm < km || (kd & (nd - kd)) != 0) && return 0
        c(nd, n) - c(kd, k) - c((n - k) ÷ 3, n - k)
    end
    [T(n, k) for k in 0:n]
end
for n in 0:12 println(T_row(n)) end  # Peter Luschny, Jul 02 2025

A385608[n_] := A385608[n] = 2*# + Quotient[n, 6] - DigitSum[#, 2] & [Quotient[n, 3]];
A385458[n_, k_] := A385608[n] - A385608[k] - A385608[n-k];
Table[A385458[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Jul 04 2025 *)

def b(n): return 2*(n//3) + n//6 - (n//3).bit_count()
def T(n, k): return b(n) - b(k) - b(n-k) # David Radcliffe, Jul 01 2025

T(n, k) = A007814(A010048(n, k)).
T(n, k) = Sum_{i=1..k} (A337923(n+1-i) - A337923(i)).
T(n, k) = b(n) - b(k) - b(n - k), where b(n) = 2*floor(n/3) + floor(n/6) - A000120(floor(n/3)) = A385608(n) is the 2-adic valuation of the product of the first n Fibonacci numbers.
sign(T(n, k)) = 1 - A385456(n, k). - Peter Luschny, Jul 03 2025

A385608 a(n) = 2-adic valuation of A003266(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 4, 5, 5, 5, 9, 9, 9, 10, 10, 10, 13, 13, 13, 14, 14, 14, 19, 19, 19, 20, 20, 20, 23, 23, 23, 24, 24, 24, 28, 28, 28, 29, 29, 29, 32, 32, 32, 33, 33, 33, 39, 39, 39, 40, 40, 40, 43, 43, 43, 44, 44, 44, 48, 48, 48, 49, 49, 49, 52, 52, 52, 53, 53, 53

Offset: 0

Paolo Xausa, Jul 04 2025

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000120, A003266, A007814, A385458, A385609.

Partial sums of A337923.

A385608[n_] := 2*# + Quotient[n, 6] - DigitSum[#, 2] & [Quotient[n, 3]];
Array[A385608, 100, 0] (* or *)
Join[{0}, Accumulate[IntegerExponent[Fibonacci[Range[99]], 2]]]

a(n) = 2*floor(n/3) + floor(n/6) - A000120(floor(n/3)) (formula by David Radcliffe at A385458).
a(n) = A007814(A003266(n)).
For n >= 1, a(n) = Sum_{k=1..n} A337923(k).
a(3*k) = a(3*k+1) = a(3*k+2), for k >= 0.

A083523 Smallest Fibonacci number divisible by 2^n.

Original entry on oeis.org

1, 2, 8, 8, 144, 46368, 4807526976, 51680708854858323072, 5972304273877744135569338397692020533504, 79757008057644623350300078764807923712509139103039448418553259155159833079730688

Offset: 0

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 05 2003

nonn

The index of the Fibonacci numbers above begin: 1, 3, 6, 6 and then doubles thereafter.

Amiram Eldar, Table of n, a(n) for n = 0..12
Ron Knott, The first 300 Fibonacci numbers, completely factorised.
Tamás Lengyel, The order of the Fibonacci and Lucas numbers, The Fibonacci Quarterly, Vol. 33, No. 3 (1995), pp. 234-239.

Cf. A000045, A001622, A007283, A079613, A337923.

Do[k = 1; While[ !IntegerQ[ Fibonacci[k]/2^n], k++ ]; Print[ Fibonacci[k]], {n, 0, 10}]
With[{fibs=Fibonacci[Range[1000]]},Table[SelectFirst[fibs, Divisible[#,2^n]&],{n,0,10}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 02 2021 *)
Join[{1, 2, 8}, Table[Fibonacci[3*2^(n - 2)], {n, 3, 9}]] (* Amiram Eldar, Jan 29 2022 *)

From Amiram Eldar, Jan 29 2022: (Start)
a(n) = Fibonacci(3*2^(n-2)) = A000045(A007283(n-2)) = A079613(n-2), for n > 2.
Sum_{n>=0} 1/a(n) = 19/8 - 1/phi, where phi is the golden ratio (A001622). (End)

Edited and extended by Robert G. Wilson v, May 06 2003

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A090740 Exponent of 2 in 3^n - 1.

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A385458 Triangle read by rows: T(n,k) = exponent of the highest power of 2 dividing each Fibonomial coefficient fibonomial(n, k).

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A385608 a(n) = 2-adic valuation of A003266(n).

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A083523 Smallest Fibonacci number divisible by 2^n.

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