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A006694
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Number of cyclotomic cosets of 2 mod 2n+1.
(Formerly M0192)
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40
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0, 1, 1, 2, 2, 1, 1, 4, 2, 1, 5, 2, 2, 3, 1, 6, 4, 5, 1, 4, 2, 3, 7, 2, 4, 7, 1, 4, 4, 1, 1, 12, 6, 1, 5, 2, 8, 7, 5, 2, 4, 1, 11, 4, 8, 9, 13, 4, 2, 7, 1, 2, 14, 1, 3, 4, 4, 5, 11, 8, 2, 7, 3, 18, 10, 1, 9, 10, 2, 1, 5, 4, 6, 9, 1, 10, 12, 13, 3, 4, 8, 1, 13, 2, 2, 11, 1, 8, 4, 1, 1, 4, 6, 7, 19, 2, 2, 19, 1, 2
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OFFSET
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0,4
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COMMENTS
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a(0) = 0 by convention.
The number of cycles in permutations constructed from siteswap juggling patterns 1, 123, 12345, 1234567, etc., i.e., the number of ball orbits in such patterns minus one.
Also the number of irreducible polynomial factors of the polynomial (x^(2n+1) - 1) / (x - 1) over GF(2). - V. Raman, Oct 04 2012
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REFERENCES
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F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977, pp. 104-105.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Ray Chandler, Table of n, a(n) for n = 0..10000
J.-P. Allouche, Suites infinies à répétitions bornées, Séminaire de Théorie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.
J.-P. Allouche, Suites infinies à répétitions bornées, Séminaire de Théorie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.
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FORMULA
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Conjecture: a((3^n-1)/2) = n. - Vladimir Shevelev, May 26 2008 [This is correct.
2*((3^n-1)/2) + 1 = 3^n and the polynomial (x^(3^n) - 1) / (x - 1) factors over GF(2) into Prod_{k=0}^{n-1} x^(2*3^k) + x^(3^k) + 1. - Joerg Arndt, Apr 01 2019]
a(n) = A081844(n) - 1.
a(n) = A064286(n) + 2*A064287(n).
From Vladimir Shevelev, Jan 19 2011: (Start)
1) A006694(n)=A037226(n) iff 2n+1 is prime;
2) The only case when A006694(n) < A037226(n) is n=0;
3) If {C_i}, i=1..A006694(n), is the set of all cyclotomic cosets of 2 mod (2n+1), then lcm(|C_1|, ..., |C_{A006694(n)}|) = A002326(n). (End)
a(n) = A000374(2*n + 1) - 1. - Joerg Arndt, Apr 01 2019
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EXAMPLE
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Mod 15 there are 4 cosets: {1, 2, 4, 8}, {3, 6, 12, 9}, {5, 10}, {7, 14, 13, 11}, so a(7) = 4. Mod 13 there is only one coset: {1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7}, so a(6) = 1.
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MAPLE
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with(group); with(numtheory); gen_rss_perm := proc(n) local a, i; a := []; for i from 1 to n do a := [op(a), ((2*i) mod (n+1))]; od; RETURN(a); end; count_of_disjcyc_seq := [seq(nops(convert(gen_rss_perm(2*j), 'disjcyc')), j=0..)];
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MATHEMATICA
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Needs["Combinatorica`"]; f[n_] := Length[ToCycles[Mod[2Range[2n], 2n + 1]]]; Table[f[n], {n, 0, 100}] (* Ray Chandler, Apr 25 2008 *)
f[n_] := Length[FactorList[x^(2n + 1) - 1, Modulus -> 2]] - 2; Table[f[n], {n, 0, 100}] (* Ray Chandler, Apr 25 2008 *)
a[n_] := Sum[ EulerPhi[d] / MultiplicativeOrder[2, d], {d, Divisors[2n + 1]}] - 1; Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Dec 14 2011, after Joerg Arndt *)
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PROG
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(PARI) a(n)=sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1; /* cf. A081844 */
vector(122, n, a(n-1)) \\ Joerg Arndt, Jan 18 2011
(PARI) vector(100, p, matsize(factormod((x^(2*p+1)+1)/(x+1), 2, 1))[1]) \\ V. Raman, Oct 04 2012
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CROSSREFS
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Cf. A002326 (order of 2 mod 2n+1), A139767.
A001917 gives cycle counts of such permutations constructed only for odd primes.
Cf. A000374 (number of factors of x^n - 1 over GF(2)).
Sequence in context: A292201 A090048 A064285 * A210481 A217209 A233307
Adjacent sequences: A006691 A006692 A006693 * A006695 A006696 A006697
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane, Sep 25 2001
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EXTENSIONS
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Additional comments from Antti Karttunen, Jan 05 2000
Extended by Ray Chandler, Apr 25 2008
Edited by N. J. A. Sloane, Apr 27 2008 at the suggestion of Ray Chandler
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STATUS
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approved
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