A090740 -id:A090740 - cp's OEIS frontend

A182000 G.f.: exp( Sum_{n>=1} 2^A090740(n) * x^n/n ) where A090740(n) = highest exponent of 2 in 3^n-1.

1, 2, 6, 10, 22, 34, 62, 90, 150, 210, 326, 442, 654, 866, 1230, 1594, 2198, 2802, 3766, 4730, 6230, 7730, 9998, 12266, 15630, 18994, 23878, 28762, 35742, 42722, 52526, 62330, 75926, 89522, 108118, 126714, 151878, 177042, 210702, 244362, 288982, 333602, 392182

Offset: 0

Paul D. Hanna, Apr 17 2012

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 22*x^4 + 34*x^5 + 62*x^6 +...
The g.f. satisfies:
A(x)/A(x^2) = 1 + 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 +...+ 2*n*x^n +...
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 16*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 32*x^8/8 + 2*x^9/9 + 8*x^10/10 + 2*x^11/11 + 16*x^12/12 +...+ 2^A090740(n)*x^n/n +...
where the highest exponents of 2 in 3^n-1, for n>=1, begins:
A090740 = [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,...].
The g.f.s of the BISECTIONS begin:
B_0(x) = 1 + 6*x + 22*x^2 + 62*x^3 + 150*x^4 + 326*x^5 + 654*x^6 +...
B_1(x) = 2 + 10*x + 34*x^2 + 90*x^3 + 210*x^4 + 442*x^5 + 866*x^6 +...
where 2 * B_0(x) / B_1(x) = 1+x.

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Cf. A173283, A161809, A182185, A090740.

{a(n)=polcoeff(exp(sum(m=1,n+1,2^valuation(3^m-1,2)*x^m/m)+x*O(x^n)),n)}
for(n=0,40,print1(a(n),", "))

{a(n)=local(A=1+x);for(i=1,#binary(n)+1,A=(1+x^2)/(1-x)^2*subst(A,x,x^2+x*O(x^n)));polcoeff(A,n)}

G.f. satisfies: A(x) = (1+x^2)/(1-x)^2 * A(x^2).

Define BISECTIONS: A(x) = B_0(x^2) + x*B_1(x^2), then: B_1(x)/B_0(x) = 2/(1+x).

A385609 Partial sums of A090740.

Original entry on oeis.org

1, 4, 5, 9, 10, 13, 14, 19, 20, 23, 24, 28, 29, 32, 33, 39, 40, 43, 44, 48, 49, 52, 53, 58, 59, 62, 63, 67, 68, 71, 72, 79, 80, 83, 84, 88, 89, 92, 93, 98, 99, 102, 103, 107, 108, 111, 112, 118, 119, 122, 123, 127, 128, 131, 132, 137, 138, 141, 142, 146, 147, 150, 151

Offset: 1

Paolo Xausa, Jul 04 2025

Prepended with 0, a trisection of A385608.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000120, A090740, A385608.

A385609[n_] := 2*n + Quotient[n, 2] - DigitSum[n, 2];
Array[A385609, 100] (* or *)
Accumulate[IntegerExponent[3^Range[100] - 1, 2]]

a(n) = Sum_{k=1..n} A090740(k).
a(n) = 2*n + floor(n/2) - A000120(n).
a(n) = A385608(n*3).

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

Original entry on oeis.org

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

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

A337923 a(n) is the exponent of the highest power of 2 dividing the n-th Fibonacci number.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 5, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 5, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1

Offset: 1

Amiram Eldar, Jan 29 2021

nonn

			a(1) = 0 since Fibonacci(1) = 1 is odd.
a(6) = 3 since Fibonacci(6) = 8 = 2^3.
a(12) = 4 since Fibonacci(12) = 144 = 2^4 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tamás Lengyel, The order of the Fibonacci and Lucas numbers, The Fibonacci Quarterly, Vol. 33, No. 3 (1995), pp. 234-239.

Cf. A000045, A007814, A248174.
Cf. A001651, A016945, A017593, A073762, A184985.
Cf. A090740 (sequence without zeros).

a[n_] := IntegerExponent[Fibonacci[n], 2]; Array[a, 100]

def A337923(n): return int(not n%3)+(int(not n%6)<<1) if n%12 else 2+(~n&n-1).bit_length() # Chai Wah Wu, Jul 10 2022

a(n) = A007814(A000045(n)).
The following 4 formulas completely specify the sequence (Lengyel, 1995):
1. a(n) = 0 if n == 1 (mod 3) or n == 2 (mod 3).
2. a(n) = 1 if n == 3 (mod 6).
3. a(n) = 3 if n == 6 (mod 12).
4. a(n) = A007814(n) + 2 if n == 0 (mod 12).
a(A001651(n)) = 0.
a(A016945(n)) = 1.
a(A017593(n)) = 3.
a(A073762(n)) = 4.
The image of this function is A184985, i.e., all the nonnegative integers excluding 2.
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 5/6.
a(3*n) = A090740(n), a(3*n+1) = a(3*n+2) = 0. - Joerg Arndt, Mar 01 2023

A060865 a(n) is the exact power of 2 that divides the n-th Fibonacci number (A000045).

Original entry on oeis.org

1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 16, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 32, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 16, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 64, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 16, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 32, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 16, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 128, 1, 1, 2, 1

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001

			a(12) = 16 because the 12th Fibonacci number is 144 and 144 = 9*16.

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

Cf. A000045, A060904(n) = 5^A112765(n), A090740.

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

Table[2^IntegerExponent[Fibonacci[n],2],{n,100}] (* Harvey P. Dale, Aug 04 2025 *)

a(n)=2^valuation(fibonacci(n), 2) \\Michel Marcus, Jul 30 2013

If n is not divisible by 3 then a(n) = 1, if n = 3 * 2^k * (2m + 1) then a(n) = 2 if k=0 or 2^(k+2) if k>0.
a(n) = F(n) / A174883(n). - Franklin T. Adams-Watters, Jan 24 2012
a(n) = A006519(A000045(n)). - Michel Marcus, Jul 30 2013
a(3n) = 2^A090740(n). - Robert Israel, Dec 28 2015

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

A093052 Exponent of 2 in 6^n - 2^n.

Original entry on oeis.org

0, 2, 5, 4, 8, 6, 9, 8, 13, 10, 13, 12, 16, 14, 17, 16, 22, 18, 21, 20, 24, 22, 25, 24, 29, 26, 29, 28, 32, 30, 33, 32, 39, 34, 37, 36, 40, 38, 41, 40, 45, 42, 45, 44, 48, 46, 49, 48, 54, 50, 53, 52, 56, 54, 57, 56, 61, 58, 61, 60, 64, 62, 65, 64, 72, 66, 69, 68, 72

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

a(n-1) is the exponent of 2 in A009168(n), A012394(n), A088991(n), A009083(n), A012036(n), A012092(n), A012395(n), A012460(n), A012465(n), A012466(n), A012467(n), (A049294(n)-1)/3.

Cf. A093050, A093051.

Join[{0},Table[IntegerExponent[6^n-2^n,2],{n,70}]] (* Harvey P. Dale, Mar 08 2012 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+2*floor((n+2)/4)+1,n+1))

def A093052(n): return n+(~(m:=3**n-1)& m-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + [(n+1)/2] + 1, a(2n+1) = 2n+2.

a(n) = n + A007814(A024023(n)) = n + A090740(n). - Reinhard Zumkeller, Mar 27 2004

A162728 G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.

Original entry on oeis.org

1, 3, 2, 8, 4, 6, 6, 20, 6, 12, 10, 16, 12, 18, 8, 48, 16, 18, 18, 32, 12, 30, 22, 40, 20, 36, 18, 48, 28, 24, 30, 112, 20, 48, 24, 48, 36, 54, 24, 80, 40, 36, 42, 80, 24, 66, 46, 96, 42, 60, 32, 96, 52, 54, 40, 120, 36, 84, 58, 64, 60, 90, 36, 256, 48, 60, 66, 128, 44, 72, 70

Offset: 1

Paul D. Hanna, Jul 12 2009

Dirichlet inverse of A117212. - R. J. Mathar, Jul 15 2010

			x/(1-x) = log(1+x) + 3*log(1+x^2)/2 + 2*log(1+x^3)/3 + 8*log(1+x^4)/4 + 4*log(1+x^5)/5 + 6*log(1+x^6)/6 + 6*log(1+x^7)/7 + 20*log(1+x^8)/8 +...

Paul D. Hanna, Table of n, a(n) for n = 1..10000

Cf. A090739, A091512, A000010 (Euler phi), A220466.

Cf. A007814, A001511, A090740, A085058, A055034, A008683.

nmax:=71: with(numtheory): 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) := (p+2)*2^(p-1)*phi(2*n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 26 2013

f[p_, e_] := (p-1)*p^(e-1); f[2, e_] := (e+2)*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

/* As the inverse Mobius transform of A091512: */
{a(n)=sumdiv(n,d,moebius(n/d)*valuation((2*d)^d,2))}

/* From a(2n-1)=phi(2n-1); a(2n)=phi(2n)*A090739(n), we get: */
{a(n)=if(n%2==1,eulerphi(n),eulerphi(n)*valuation(3^n-1,2))}

/* From x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n, we get: */
{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0);A[ #A]=#A*(1-polcoeff(sum(m=1,#A,A[m]/m*log(1+x^m +x*O(x^#A)) ),#A)));A[n]}

a(2n-1) = phi(2n-1); a(2n) = phi(2n)*A090739(n), where A090739(n) = exponent of 2 in 3^(2n)-1.
Inverse Mobius transform of A091512, where A091512(n) = exponent of 2 in (2n)^n.
Multiplicative: a(m*n) = a(m)*a(n) when gcd(m,n)=1, with a(p) = p-1 for odd prime p and a(2)=3.
G.f.: x/(1-x)^2 = Sum_{n>=1} a(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
Dirichlet g.f.: zeta(s-1)/( zeta(s)*(1-2^(1-s)) ). - R. J. Mathar, Apr 14 2011
a((2*n-1)*2^p) = (p+2)*2^(p-1)* phi(2*n-1), p >= 0. Observe that a(2^p) = A001792(p). - Johannes W. Meijer, Jan 26 2013
Sum_{k=1..n} a(k) ~ 6*n^2 / Pi^2. - Vaclav Kotesovec, Feb 07 2019
Multiplicative with a(2^e) = (e+2)*2^(e-1) and a(p^e) = (p-1)*p^(e-1) for an odd prime p. - Amiram Eldar, Aug 27 2023
From Ridouane Oudra, Jul 05 2025: (Start)
a(n) = Sum_{k=0..A007814(n)} 2^k*phi(n/2^k).
a(n) = Sum_{d|n} mu(n/d)*d*A001511(d).
a(n) = A000010(n)*A090740(n).
a(n) = A085058(n-1)*A055034(n), for n>1. (End)

A091283 Exponent of 2 in -1+prime[n]^s, if s is an exponent of the form s=8k-4.

Original entry on oeis.org

0, 4, 4, 5, 4, 4, 6, 4, 5, 4, 7, 4, 5, 4, 6, 4, 4, 4, 4, 5, 5, 6, 4, 5, 7, 4, 5, 4, 4, 6, 9, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 8, 8, 4, 5, 4, 7, 4, 4, 5, 6, 6, 4, 10, 5, 4, 6, 4, 5, 4, 4, 4, 5, 5, 4, 4, 6, 4, 4, 7, 5, 6, 4, 4, 9, 4, 4, 6, 5, 4, 4, 6, 6, 5, 4, 8, 5, 4, 6, 4, 7, 5, 4, 4, 5, 4, 5, 4, 4, 4, 4, 4, 5, 4

Offset: 1

Labos Elemer, Jan 22 2004

nonn

Cf. A091282, A090739, A090740, A090129, A023506.

Table[{8*k-4, Table[Part[Flatten[FactorInteger [ -1+Prime[n]^(8*k-4)]], 2], {n, 2, m}]}, {k, 1, 2}]

cp's OEIS Frontend

A182000 G.f.: exp( Sum_{n>=1} 2^A090740(n) * x^n/n ) where A090740(n) = highest exponent of 2 in 3^n-1.

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A385609 Partial sums of A090740.

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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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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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A337923 a(n) is the exponent of the highest power of 2 dividing the n-th Fibonacci number.

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A060865 a(n) is the exact power of 2 that divides the n-th Fibonacci number (A000045).

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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.