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 A006694 Number of cyclotomic cosets of 2 mod 2n+1. (Formerly M0192) 39

%I M0192

%S 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,

%T 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,

%U 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

%N Number of cyclotomic cosets of 2 mod 2n+1.

%C a(0) = 0 by convention.

%C 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.

%C Also the number of irreducible polynomial factors of the polynomial (x^(2n+1) - 1) / (x - 1) over GF(2). - _V. Raman_, Oct 04 2012

%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977, pp. 104-105.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ray Chandler, <a href="/A006694/b006694.txt">Table of n, a(n) for n = 0..10000</a>

%H J.-P. Allouche, <a href="http://www.jstor.org/stable/44165489">Suites infinies à répétitions bornées</a>, Séminaire de Théorie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.

%H J.-P. Allouche, <a href="/A006694/a006694.pdf">Suites infinies à répétitions bornées</a>, Séminaire de Théorie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.

%F Conjecture: a((3^n-1)/2) = n. - _Vladimir Shevelev_, May 26 2008 [This is correct.

%F 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]

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

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

%F From _Vladimir Shevelev_, Jan 19 2011: (Start)

%F 1) A006694(n)=A037226(n) iff 2n+1 is prime;

%F 2) The only case when A006694(n) < A037226(n) is n=0;

%F 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)

%F a(n) = A000374(2*n + 1) - 1. - _Joerg Arndt_, Apr 01 2019

%e 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.

%p 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..)];

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

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

%t 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_ *)

%o (PARI) a(n)=sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1; /* cf. A081844 */

%o vector(122, n, a(n-1)) \\ _Joerg Arndt_, Jan 18 2011

%o (PARI) vector(100, p, matsize(factormod((x^(2*p+1)+1)/(x+1), 2, 1))[1]) \\ _V. Raman_, Oct 04 2012

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

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

%Y Cf. A000374 (number of factors of x^n - 1 over GF(2)).

%K nonn,nice,easy

%O 0,4

%A _N. J. A. Sloane_, Sep 25 2001

%E Additional comments from _Antti Karttunen_, Jan 05 2000

%E Extended by _Ray Chandler_, Apr 25 2008

%E Edited by _N. J. A. Sloane_, Apr 27 2008 at the suggestion of _Ray Chandler_

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Last modified October 22 09:56 EDT 2019. Contains 328315 sequences. (Running on oeis4.)