

A090129


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


18



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
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OFFSET

1,2


COMMENTS

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
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



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.
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

(Python)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

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


STATUS

approved



