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A023135
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Number of cycles of function f(x) = 3x mod n.
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10
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| Number of factors in the factorization of the polynomial x^n-1 over the integers mod 3. - T. D. Noe (noe(AT)sspectra.com), Apr 16 2003
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REFERENCES
| R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, p. 65.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
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FORMULA
| a(n) = Sum_{d|m} phi(d)/ord(3, d), where m is n with all factors of 3 removed. - T. D. Noe (noe(AT)sspectra.com), Apr 19 2003
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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.
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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}]
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CROSSREFS
| Cf. A000005, A000374, A023136-A023142.
Sequence in context: A035191 A177062 A133924 * A191654 A205784 A066272
Adjacent sequences: A023132 A023133 A023134 * A023136 A023137 A023138
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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