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A000374
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Number of cycles (mod n) under doubling map.
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20
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1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 2, 3, 5, 1, 3, 3, 2, 2, 6, 2, 3, 2, 3, 2, 4, 3, 2, 5, 7, 1, 5, 3, 6, 3, 2, 2, 5, 2, 3, 6, 4, 2, 8, 3, 3, 2, 5, 3, 8, 2, 2, 4, 5, 3, 5, 2, 2, 5, 2, 7, 13, 1, 7, 5, 2, 3, 6, 6, 3, 3, 9, 2, 8, 2, 6, 5, 3, 2, 5, 3, 2, 6, 12, 4, 5, 2, 9, 8, 10, 3, 14, 3, 5, 2, 3, 5, 8, 3
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OFFSET
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1,3
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COMMENTS
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Number of cycles of the function f(x) = 2x mod n. Number of irreducible factors in the factorization of the polynomial x^n-1 over the integers mod 2. - T. D. Noe, 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
Jarkko Peltomäki and Aleksi Saarela, Standard words and solutions of the word equation X_1^2 ... X_n^2 = (X_1 ... X_n)^2, Journal of Combinatorial Theory, Series A (2021) Vol. 178, 105340. See also arXiv:2004.14657 [cs.FL], 2020.
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FORMULA
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a(n) = Sum_{d|m} phi(d)/ord(2, d), where m is n with all factors of 2 removed. - T. D. Noe, Apr 19 2003
a(n) = (1/ord(2,m))*Sum_{j = 0..ord(2,m)-1} gcd(2^j - 1, m), where m is n with all factors of 2 removed. - Nihar Prakash Gargava, Nov 12 2018
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EXAMPLE
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a(14) = 3 because (1) the function 2x mod 14 has the three cycles (0),(2,4,8),(6,12,10) and (2) the factorization of x^14-1 over integers mod 2 is (1+x)^2 (1+x+x^3)^2 (1+x^2+x^3)^2, which has three unique factors. Note that the length of the cycles is the same as the degree of the factors.
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MATHEMATICA
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Table[Length[FactorList[x^n - 1, Modulus -> 2]] - 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[2, n], {n, 100}]
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PROG
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(PARI) a(n)={sumdiv(n >> valuation(n, 2), d, eulerphi(d)/znorder(Mod(2, d)))} \\ Andrew Howroyd, Nov 12 2018
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CROSSREFS
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Cf. A000005, A023135-A023142.
Cf. A081844 (number of irreducible factors of x^(2n+1) - 1 over GF(2)).
Cf. A037226 (number of primitive irreducible factors of x^(2n+1) - 1 over integers mod 2).
Sequence in context: A261787 A302480 A329656 * A355735 A355733 A256757
Adjacent sequences: A000371 A000372 A000373 * A000375 A000376 A000377
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KEYWORD
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nonn
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AUTHOR
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Shel Kaphan
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STATUS
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approved
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