login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A023135 Number of cycles of function f(x) = 3x mod n. 10
1, 2, 1, 3, 2, 2, 2, 5, 1, 4, 3, 3, 5, 4, 2, 7, 2, 2, 2, 7, 2, 6, 3, 5, 3, 10, 1, 7, 2, 4, 2, 9, 3, 4, 5, 3, 3, 4, 5, 13, 6, 4, 2, 9, 2, 6, 3, 7, 3, 6, 2, 15, 2, 2, 6, 13, 2, 4, 3, 7, 7, 4, 2, 11, 10, 6, 4, 7, 3, 10, 3, 5, 7, 6, 3, 7, 6, 10, 2, 23, 1, 12, 3, 7, 7, 4, 2, 15, 2, 4, 18, 9, 2, 6, 5, 9, 3, 6, 3, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of factors in the factorization of the polynomial x^n-1 over the integers mod 3. - T. D. Noe, Apr 16 2003

REFERENCES

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, p. 65.

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

FORMULA

a(n) = Sum_{d|m} phi(d)/ord(3, d), where m is n with all factors of 3 removed. - T. D. Noe, Apr 19 2003

EXAMPLE

a(15) = 2 because (1) the function 3x mod 15 has the two cycles (0),(3,9,12,6) and (2) the factorization of x^15-1 over integers mod 3 is (2+x)^3 (1+x+x^2+x^3+x^4)^3, which has two unique factors. Note that the length of the cycles is the same as the degree of the factors.

MATHEMATICA

Table[Length[FactorList[x^n - 1, Modulus -> 3]] - 1, {n, 100}]

CountFactors[p_, n_] := Module[{sum=0, m=n, d, f, i}, While[Mod[m, p]==0, m/=p]; d=Divisors[m]; Do[f=d[[i]]; sum+=EulerPhi[f]/MultiplicativeOrder[p, f], {i, Length[d]}]; sum]; Table[CountFactors[3, n], {n, 100}]

CROSSREFS

Cf. A000005, A000374, A023136-A023142.

Sequence in context: A276166 A177062 A133924 * A191654 A205784 A066272

Adjacent sequences:  A023132 A023133 A023134 * A023136 A023137 A023138

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 19 05:39 EDT 2018. Contains 316336 sequences. (Running on oeis4.)