

A090129


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


14



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
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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^(n2).
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 (41)^(2^(k2)) 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^(n1), 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^(n2) 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 noncyclic (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. A068531, A069895, A088660, A090739, A090740, A091512.
Sequence in context: A286496 A318187 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



