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Search: id:A023139
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| A023139 |
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Number of cycles of function f(x) = 7x mod n. |
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1
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| 1, 2, 3, 3, 2, 6, 1, 5, 5, 4, 2, 9, 2, 2, 6, 9, 2, 10, 7, 7, 3, 4, 2, 15, 7, 4, 7, 3, 5, 12, 3, 13, 6, 4, 2, 15, 5, 14, 6, 13, 2, 6, 8, 7, 10, 4, 3, 27, 1, 14, 6, 7, 3, 14, 5, 5, 21, 10, 3, 21, 2, 6, 5, 17, 7, 12, 2, 7, 6, 4, 2, 25, 4, 10, 21, 21, 2, 12, 2, 25, 9, 4, 3, 9, 7, 16, 15, 13, 2, 20, 2, 7, 9, 6
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Number of factors in the factorization of the polynomial x^n-1 over the integers mod 7. - T. D. Noe (noe(AT)sspectra.com), Apr 16 2003
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REFERENCES
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R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, p. 65.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
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FORMULA
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a(n) = Sum_{d|m} phi(d)/ord(7, d), where m is n with all factors of 7 removed. - T. D. Noe (noe(AT)sspectra.com), Apr 19 2003
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EXAMPLE
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a(8) = 5 because (1) the function 7x mod 8 has the five cycles (0),(4),(1,7),(2,6),(3,5) and (2) the factorization of x^8-1 over integers mod 7 is (1+x) (6+x) (1+x^2) (1+3x+x^2) (1+4x+x^2), which has five unique factors. Note that the length of the cycles is the same as the degree of the factors.
a(10) = 2 because the function 8x mod 10 has the two cycles (0),(2,6,8,4).
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MATHEMATICA
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Table[Length[FactorList[x^n - 1, Modulus -> 7]] - 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[7, n], {n, 100}]
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CROSSREFS
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Cf. A000005, A000374, A023135-A023138, A023140-A023142.
Cf. A000374, A023135-A023142.
Sequence in context: A095174 A131307 A091813 this_sequence A153290 A153516 A153491
Adjacent sequences: A023136 A023137 A023138 this_sequence A023140 A023141 A023142
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson (davidwwilson(AT)comcast.net)
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