A069895 -id:A069895 - cp's OEIS frontend

A090129 Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.

1, 2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184

Offset: 1

Labos Elemer and Ralf Stephan, Jan 19 2004

A131577 and A011782 are companions, A131577(n) + A011782(n) = 2^n, (and differences each other). - Paul Curtz, Jan 18 2009
A090127 with offset 0: (1, 2, 2, 4, 8, ...) = A(x) / A(x^2), when A(x) = (1 + 2x + 4x^2 + 8x^3 + ...). - Gary W. Adamson, Feb 20 2010
From Wolfdieter Lang, Apr 18 2012: (Start)
a(n) is the order of 3 modulo 2^n. For n=1 and 2 this is obviously 1 and 2, respectively, and for n >= 3 it is 2^(n-2).
For a proof see, e.g., the Graeme McRae link under A068531, the section 'A Different Approach', proposed by Alexander Monnas, the first part, where the result from the expansion of (4-1)^(2^(k-2)) holds only for k >= 3. See also the Charles R Greathouse IV program below where this result has been used.
This means that the cycle generated by 3, taken modulo 2^n, has length a(n), and that 3 is not a primitive root modulo 2^n, if n >= 3 (because Euler's phi(2^n) = 2^(n-1), n >= 1, see A000010).
(End)
Let r(x) = (1 + 2x + 2x^2 + 4x^3 + ...). Then (1 + 2x + 4x^2 + 8x^3 + ...) = (r(x) * r(x^2) * r(x)^4 * r(x^8) * ...). - Gary W. Adamson, Sep 13 2016

			a(1) = 1 since -1 + 3 = 2 is divisible by 2^1;
a(2) = a(3) = 2 since -1 + 9 = 8 is divisible by 4 = 2^2 and also by 8 = 2^3;
a(5) = 8 since -1 + 6561 = 6560 = 32*205 is divisible by 2^5.
From _Wolfdieter Lang_, Apr 18 2012: (Start)
n=3: the order of 3 (mod 8) is a(3)=2 because the cycle generated by 3 is [3, 3^2==1 (mod 8)].
n=5: a(5) = 2^3 = 8 because the cycle generated by 3 is [3^1=3, 3^2=9, 3^3=27, 17, 19, 25, 11, 1] (mod 32).
  The multiplicative group mod 32 is non-cyclic (see A033949(10)) with the additional four cycles  [5, 25, 29, 17, 21, 9, 13, 1], [7, 17, 23, 1], [15, 1], and [31, 1]. This is the cycle structure of the (Abelian) group Z_8 x Z_2 (see one of the cycle graphs shown in the Wikipedia link 'List of small groups' for the order phi(32)=16, given under A192005).
(End)

Index to divisibility sequences

Cf. A068531, A069895, A088660, A090739, A090740, A091512.

Essentially the same as A000079.

t=Table[Part[Flatten[FactorInteger[ -1+3^(n)]], 2], {n, 1, 130}] Table[Min[Flatten[Position[t, j]]], {j, 1, 10}]
Join[{1,2},2^Range[30]] (* or *) Join[{1,2},NestList[2#&,2,30]] (* Harvey P. Dale, Nov 08 2012 *)

a(n)=2^(n+(n<3)-2) \\ Charles R Greathouse IV, Apr 09 2012

def A090129(n): return n if n<3 else 1<Chai Wah Wu, Jul 11 2022

a(n) = 2^(n-2) if n >= 3, 1 for n=1 and 2 for n=2 (see the order comment above).

a(n+2) = A152046(n) + A152046(n+1) = 2*A011782(n). - Paul Curtz, Jan 18 2009

a(11) through a(20) from R. J. Mathar, Aug 08 2008

More terms (powers of 2, see a comment above) from Wolfdieter Lang, Apr 18 2012

A090739 Exponent of 2 in 9^n - 1.

Original entry on oeis.org

3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 8, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 9, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3

Offset: 1

Labos Elemer and Ralf Stephan, Jan 19 2004

The exponent of 2 in the factorization of Fibonacci(6n). - T. D. Noe, Mar 14 2014

Records of 3, 4, 5, 6, 7, 8,.. occur at n= 1, 2, 4, 8, 16, 32,... - R. J. Mathar, Jun 28 2025

			For n = 2, we see that -1 + 3^4 = 80 = 2^4 * 5 so a(2) = 4.
For n = 3, we see that -1 + 3^6 = 728 = 2^3 * 7 * 13, so a(3) = 3.

T. D. Noe, Table of n, a(n) for n = 1..1000
T. Lengyel, The order of the Fibonacci and Lucas numbers, Fib. Quart. 33 (1995), 234-239.

Cf. A024101, A069895, A091512, A088660, A090740, A220466.
Cf. A000005, A006519, A120738 (partial sums).
Appears in A161737.

A090739 := proc(n)
    padic[ordp](9^n-1,2) ;
end proc:
seq(A090739(n),n=1..80) ; # R. J. Mathar, Jun 28 2025

Table[Part[Flatten[FactorInteger[ -1+3^(2*n)]], 2], {n, 1, 70}]
Table[IntegerExponent[Fibonacci[n], 2], {n, 6, 600, 6}] (* T. D. Noe, Mar 14 2014 *)

a(n)=valuation(n,2)+3 \\ Charles R Greathouse IV, Mar 14 2014

def A090739(n): return (~n&n-1).bit_length()+3 # Chai Wah Wu, Jul 11 2022

a(n) = A007814(n) + 3.
a((2*n-1)*2^p) = p + 3, p >= 0. - Johannes W. Meijer, Feb 08 2013
a(n) = log_2(A006519(9^n - 1)). - Alonso del Arte, Feb 08 2013
a(n) = 2*tau(4*n)/(tau(4*n) - tau(n)), where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Nov 28 2022

More terms from T. D. Noe, Mar 14 2014

A090740 Exponent of 2 in 3^n - 1.

Original entry on oeis.org

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

A091512 a(n) is the largest integer m such that 2^m divides (2n)^n, i.e., the exponent of 2 in (2n)^n.

Original entry on oeis.org

1, 4, 3, 12, 5, 12, 7, 32, 9, 20, 11, 36, 13, 28, 15, 80, 17, 36, 19, 60, 21, 44, 23, 96, 25, 52, 27, 84, 29, 60, 31, 192, 33, 68, 35, 108, 37, 76, 39, 160, 41, 84, 43, 132, 45, 92, 47, 240, 49, 100, 51, 156, 53, 108, 55, 224, 57, 116, 59, 180, 61, 124, 63

Offset: 1

Ralf Stephan and Labos Elemer, Jan 18 2004

n times one more than the trailing 0's in the binary representation of n. - Ralf Stephan, Aug 22 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n.

Cf. A091519, A090740, A090739, A048298, A162728, A220466.

[n*(Valuation(n, 2)+1): n in [1..80]]; // Vincenzo Librandi, May 16 2013

nmax:=63: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (2*n-1)*(p+1)*2^p od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 08 2013
# second Maple program:
a:= n-> n*padic[ordp](2*n, 2):
seq(a(n), n=1..63);  # Alois P. Heinz, Jan 02 2024

Table[ Part[ Flatten[ FactorInteger[(2 n)^n]], 2], {n, 1, 124}]
Table[IntegerExponent[(2n)^n,2],{n,70}] (* Harvey P. Dale, Sep 11 2015 *)

PARI
```
a(n)=n*(valuation(n,2)+1)
```
PARI
```
a(n)=if(n<1,0,if(n%2==0,2*a(n/2)+n,n))
```

def A091512(n): return n*(n&-n).bit_length() # Chai Wah Wu, Jul 11 2022

a(n) = A007814(A000312(n)) = n*A001511(n) = A069895(n)/2.
G.f.: Sum_{k>=0} 2^k*x^2^k/(1-x^2^k)^2.
Recurrence: a(0) = 0, a(2*n) = 2*a(n) + 2*n, a(2*n+1) = 2*n+1.
Dirichlet g.f.: zeta(s-1)*2^s/(2^s-2). - Ralf Stephan, Jun 17 2007
Mobius transform of A162728, where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
a(n) = A162728(2*n)/phi(2*n), where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
a((2*n-1)*2^p) = (2*n-1)*(p+1)*2^p, p >= 0. Observe that a(2^p) = A001787(p+1). - Johannes W. Meijer, Feb 08 2013
Sum_{k=1..n} a(k) ~ n^2. - Amiram Eldar, Oct 22 2022
a(n) = Sum_{d divides n} d*A048298(n/d); that is, a(n) is the Dirichlet product of A048298(n) and A000027(n). - Peter Bala, Jan 02 2024

A249153 Exponent of 2 in the hyperfactorial of 2n: a(n) = A007814(A002109(2n)).

Original entry on oeis.org

0, 2, 10, 16, 40, 50, 74, 88, 152, 170, 210, 232, 304, 330, 386, 416, 576, 610, 682, 720, 840, 882, 970, 1016, 1208, 1258, 1362, 1416, 1584, 1642, 1762, 1824, 2208, 2274, 2410, 2480, 2696, 2770, 2922, 3000, 3320, 3402, 3570, 3656, 3920, 4010, 4194, 4288, 4768, 4866, 5066, 5168, 5480, 5586, 5802, 5912

Offset: 0

Antti Karttunen, Oct 25 2014

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Bisection of A249152.

Cf. A002109, A007814, A143157, A069895 (first differences).

Table[IntegerExponent[Hyperfactorial[2*n], 2], {n, 0, 55}] (* Amiram Eldar, Sep 10 2024 *)

from sympy import multiplicity
A249153_list, n = [0], 0
for i in range(2,20002,2):
    n += multiplicity(2,i)*i
    A249153_list.append(n) # Chai Wah Wu, Aug 21 2015

a(n) = A249152(2*n) = A007814(A002109(2*n)).
a(n) = 2*A143157(n).
a(n) ~ 2*n^2. - Amiram Eldar, Sep 10 2024

A084432 Expansion of 2/(1-x) + Sum_{k>=0} t^2(3-t)/(1+t)/(1-t)^2, where t=x^2^k.

Original entry on oeis.org

2, 5, 4, 10, 6, 11, 8, 19, 10, 17, 12, 24, 14, 23, 16, 36, 18, 29, 20, 38, 22, 35, 24, 49, 26, 41, 28, 52, 30, 47, 32, 69, 34, 53, 36, 66, 38, 59, 40, 79, 42, 65, 44, 80, 46, 71, 48, 98, 50, 77, 52, 94, 54, 83, 56, 109, 58, 89, 60, 108, 62, 95, 64, 134, 66, 101, 68, 122, 70, 107, 72, 139, 74, 113

Offset: 1

Ralf Stephan, Jun 27 2003

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A069895, A373184, A373185.

for(n=1, 100, l=ceil(log(n)/log(2)); t=polcoeff(sum(k=0, l, 1/(1-x^2^k)^2) + O(x^(n+1)), n); print1(t", "))

a(1)=2, a(2*n) = a(n)+2*n+1, a(2*n+1) = 2*n+2.
Dirichlet g.f.: 2^s/(2^s-1) * (zeta(s)+zeta(s-1)). - Ralf Stephan, Jun 17 2007
From Seiichi Manyama, May 27 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x)^2 - 1 + A(x^2).
G.f.: A(x) = Sum_{k>=0} (1/(1 - x^(2^k))^2 - 1). (End)
Conjecture: a(n) is the number of integer solutions (x,y,z) to the Diophantine equation 2^x*(y+z) = n, where 0 <= x,y,z <= n. - Joseph M. Shunia, Aug 27 2024
Sum_{k=1..n} a(k) ~ 2*n^2/3 + 2*n. - Vaclav Kotesovec, Aug 28 2024

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A090129 Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.

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A090739 Exponent of 2 in 9^n - 1.

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A090740 Exponent of 2 in 3^n - 1.

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A091512 a(n) is the largest integer m such that 2^m divides (2*n)^n, i.e., the exponent of 2 in (2*n)^n.

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A249153 Exponent of 2 in the hyperfactorial of 2n: a(n) = A007814(A002109(2n)).

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A084432 Expansion of 2/(1-x) + Sum_{k>=0} t^2(3-t)/(1+t)/(1-t)^2, where t=x^2^k.

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A091512 a(n) is the largest integer m such that 2^m divides (2n)^n, i.e., the exponent of 2 in (2n)^n.