|
| |
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
A090127with 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^(n-2).
For a proof see, e.g., the Graeme McRae link underA068531,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, seeA000010).
(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^(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.
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 (seeA033949(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 underA192005).
(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) fromWolfdieter Lang,Apr 18 2012
|
|
|
STATUS
|
approved
|
| |
|
|
