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 A006694 Number of cyclotomic cosets of 2 mod 2n+1. (Formerly M0192) 41
 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 of the polynomial (x^(2n+1) - 1) / (x - 1) over GF(2). - V. Raman, Oct 04 2012 Also, a(n) is the number of cycles of the Josephus permutation for n elements and a count of 2. For n >= 1, the Josephus permutation is given by the n-th row of A321298. See Knuth 1997 (exercise 1.3.3-29). - Pontus von Brömssen, Sep 18 2022 REFERENCES Donald E. Knuth, The Art of Computer Programming, Vol. 1, 3rd edition, Addison-Wesley, 1997. 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 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. FORMULA 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 Product_{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) a(n)=A037226(n) iff 2n+1 is prime; 2) The only case when a(n) < A037226(n) is n=0; 3) If {C_i}, i=1..a(n), is the set of all cyclotomic cosets of 2 mod (2n+1), then lcm(|C_1|, ..., |C_{a(n)}|) = A002326(n). (End) a(n) = A000374(2*n + 1) - 1. - Joerg Arndt, Apr 01 2019 a(n) = (Sum_{d|(2n+1)} phi(d)/ord(2,d)) - 1, where phi = A000010 and ord(2,d) is the multiplicative order of 2 modulo d. - Jianing Song, Nov 13 2021 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, 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 *) 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))) \\ V. Raman, Oct 04 2012 CROSSREFS Cf. A000010, A000374 (number of factors of x^n - 1 over GF(2)), A002326 (order of 2 mod 2n+1), A037226, A064286, A064287, A081844, A139767, A321298. A001917 gives cycle counts of such permutations constructed only for odd primes. Second column of A357217. Sequence in context: A343070 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 October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)