

A023135


Number of cycles of function f(x) = 3x mod n.


11



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
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OFFSET

1,2


COMMENTS

Number of factors in the factorization of the polynomial x^n1 over the integers mod 3.  T. D. Noe, Apr 16 2003


REFERENCES

R. Lidl and H. Niederreiter, Finite Fields, AddisonWesley, 1983, p. 65.


LINKS

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


FORMULA

a(n) = Sum_{dm} phi(d)/ord(3, d), where m is n with all factors of 3 removed.  T. D. Noe, Apr 19 2003
a(n) = (1/ord(3,m))*Sum_{j = 0..ord(3,m)1} gcd(3^j  1, m), where m is n with all factors of 3 removed.  Nihar Prakash Gargava, Nov 14 2018


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^151 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.
Cf. A023136, A023137, A023138, A023139, A023140, A023141, A023142.
Sequence in context: A276166 A177062 A133924 * A191654 A327983 A205784
Adjacent sequences: A023132 A023133 A023134 * A023136 A023137 A023138


KEYWORD

nonn


AUTHOR

David W. Wilson


STATUS

approved



