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A006694 Number of cyclotomic cosets of 2 mod 2n+1.
(Formerly M0192)
35
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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 for the polynomial (x^(2n+1)+1)/(x+1) over GF(2). - V. Raman, Oct 04 2012

REFERENCES

J.-P. Allouche, Suites infinies a repetitions bornees, S\'{e}minaire de Th\'{e}orie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.

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).

LINKS

Ray Chandler, Table of n, a(n) for n=0..10000

FORMULA

Conjecture: a((3^n-1)/2) = n. - Vladimir Shevelev, May 26 2008

a(n) = A081844(n) - 1.

a(n) = A064286(n) + 2*A064287(n).

From Vladimir Shevelev, Jan 19 2011: 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).

EXAMPLE

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.

MAPLE

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..)];

MATHEMATICA

Needs["Combinatorica`"]; f[n_] := Length[ToCycles[Mod[2Range[2n], 2n + 1]]]; Table[f[n], {n, 0, 100}] (* Ray Chandler *)

f[n_] := Length[FactorList[x^(2n + 1) - 1, Modulus -> 2]] - 2; Table[f[n], {n, 0, 100}] (* Ray Chandler *)

a[n_] := Sum[ EulerPhi[d] / MultiplicativeOrder[2, d], {d, Divisors[2n + 1]}] - 1; Table[a[n], {n, 0, 99}] (* Jean-Fran├žois Alcover, Dec 14 2011, after Joerg Arndt *)

PROG

(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

CROSSREFS

Cf. A002326 (order of 2 mod 2n+1), A139767.

A001917 gives cycle counts of such permutations constructed only for odd primes.

Sequence in context: A232504 A090048 A064285 * A210481 A217209 A233307

Adjacent sequences:  A006691 A006692 A006693 * A006695 A006696 A006697

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Sep 25 2001

EXTENSIONS

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

STATUS

approved

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Last modified December 19 06:44 EST 2014. Contains 252177 sequences.