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 A000198 Largest order of automorphism group of a tournament with n nodes. (Formerly M2280 N0902) 1
 1, 1, 3, 3, 5, 9, 21, 21, 81, 81, 81, 243, 243, 441, 1215, 1701, 1701, 6561, 6561, 6561, 45927, 45927, 45927, 137781, 137781, 229635, 1594323, 1594323, 1594323, 4782969, 4782969, 7971615, 14348907, 33480783, 33480783, 129140163, 129140163, 129140163 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS It appears that all terms except a(5) = 5 are divisible by a power of 3. - Jonathan Vos Post, Apr 20 2011 REFERENCES J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 81. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Joseph Myers, Table of n, a(n) for n = 1..1000 B. Alspach, A combinatorial proof of a conjecture of Goldberg and Moon, Canad. Math. Bull. 11 (1968), 655-661. B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. [Annotated copy] See g(n) as defined on page 317 (NOT on page 322). B. Alspach and J. L. Berggren, On the determination of the maximum order of the group of a tournament, Canad. Math. Bull 16 (1973), 11-14. J. D. Dixon, The maximum order of the group of a tournament, Canad. Math. Bull. 10 (1967), 503-505. FORMULA a(3^k) = 3^((3^k - 1)/2), a(5*3^k) = 5*3^((5*3^k - 5)/2), a(7*3^k) = 7*3^((7*3^k - 5)/2), and, for all other n, a(n) = max (a(i)a(n-i)) where the max is taken over 1 <= i <= n-1 (from Alspach and Berggren (1973) Theorem 4). a(3r) = (3^r)a(r), a(n) = a(n-1) for n = 1, 2 or 4 mod 9, a(9k+8) = max(a(9k+7), a(5)a(9k+3)), a(9k+5) = max(a(2)a(9k+3), a(5)a(9k), a(7)a(9k-2)), a(9k+7) = a(7)a(9k) (from Alspach and Berggren (1973) Theorem 5). MAPLE a:= proc(n) local t, r; t:= irem(n, 9);    `if`(3^ilog[3](n)=n, 3^((3^ilog[3](n)-1)/2),    `if`(irem(n, 5, 'r')=0 and 3^ilog[3](r)=r, 5*3^((5*3^ilog[3](r)-5)/2),    `if`(irem(n, 7, 'r')=0 and 3^ilog[3](r)=r, 7*3^((7*3^ilog[3](r)-5)/2),    `if`(irem(n, 3, 'r')=0, 3^r*a(r),    `if`(t in {1, 2, 4}, a(n-1),    `if`(t = 8, max(a(n-1), a(5)*a(n-5)),    `if`(t = 5, max(a(2)*a(n-2), a(5)*a(n-5), a(7)*a(n-7)),         a(7)*a(n-7) )))))))     end: seq(a(n), n=1..50);  # Alois P. Heinz, Jun 29 2012 MATHEMATICA a[n_] := a[n] = With[{t = Mod[n, 9]}, Which[ IntegerQ[Log[3, n]], 3^((1/2)*(n-1)), {q, r} = QuotientRemainder[n, 5]; r == 0 && IntegerQ[Log[3, q]], 5*3^((1/2)*(n-5)), {q, r} = QuotientRemainder[n, 7]; r == 0 && IntegerQ[Log[3, q]], 7*3^((1/2)*(n-5)), {q, r} = QuotientRemainder[n, 3]; r == 0, 3^q*a[q], MemberQ[{1, 2, 4}, t], a[n-1], t == 8, Max[a[n-1], a[5]*a[n-5]], t == 5, Max[a[2]*a[n-2], a[5]*a[n-5], a[7]*a[n-7]], True, a[7]*a[n-7]]]; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Nov 12 2012, after Alois P. Heinz *) CROSSREFS Sequence in context: A209083 A137202 A146926 * A202674 A027170 A132775 Adjacent sequences:  A000195 A000196 A000197 * A000199 A000200 A000201 KEYWORD nonn,nice AUTHOR EXTENSIONS Edited and extended by Joseph Myers, Jun 28 2012 STATUS approved

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