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A090129 Smallest exponent such that -1+3^a(n) is divisible by 2^n. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n+2) = A152046(n)+A152046(n+1) = 2*A011782(n). 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

LINKS

Table of n, a(n) for n=1..30.

Index to divisibility sequences

FORMULA

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

EXAMPLE

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)

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A171648 A189914 A217931 * A001137 A123593 A122748

Adjacent sequences:  A090126 A090127 A090128 * A090130 A090131 A090132

KEYWORD

nonn,easy

AUTHOR

Labos Elemer and Ralf Stephan, Jan 19 2004

EXTENSIONS

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

STATUS

approved

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Last modified June 25 00:41 EDT 2017. Contains 288708 sequences.