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A001676
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Number of h-cobordism classes of smooth homotopy n-spheres.
(Formerly M5197 N2261)
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8
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1, 1, 1, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24, 8, 4, 69524373504, 2, 4, 12, 67100672, 2, 3, 3, 7767211311104, 8, 32, 32, 3014494287036416, 6, 24, 120, 2303837503821447168, 192, 32, 96, 341653284209033216, 8, 11520, 48, 798366828940770681028608, 32, 12, 24, 11852230872517975212032, 24, 32, 8, 91678339751618435453288448, 2, 16, 4, 1986677733776616536315084668928, 4, 1, 24, 142211872163171481167115958878208
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OFFSET
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1,7
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COMMENTS
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For n not equal to 4 (and possibly for all n) this is the number of oriented diffeomorphism classes of differentiable structures on the n-sphere.
a(3) = 1 follows now that the Poincare conjecture has been proved.
a(n) for n != 4 is the order of S_n, the n-th group in Tables 1 and 2 (explained in Further Details p. 807) of Milnor 2011.
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REFERENCES
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M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I. Ann. of Math. (2) 77 1963 504-537.
S. O. Kochman, Stable homotopy groups of spheres. A computer-assisted approach. Lecture Notes in Mathematics, 1423. Springer-Verlag, Berlin, 1990. 330 pp. ISBN: 3-540-52468-1. [Math. Rev. 91j:55016]
S. O. Kochman and M. E. Mahowald, On the computation of stable stems. The Cech Centennial (Boston, MA, 1993), 299-316, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. [Math. Rev. 96j:55018]
J. P. Levine, Lectures on groups of homotopy spheres. In Algebraic and geometric topology (New Brunswick, NJ, 1983), 62-95, Lecture Notes in Math., 1126, Springer, Berlin, 1985.
J. W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. 64 (1956), 399-405.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
S. P. Novikov ed., Topology I, Encyc. of Math. Sci., vol. 12.
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).
H. Whitney, The work of John W. Milnor, pp. 48-50 of Proc. Internat. Congress Mathematicians, Stockholm, 1962.
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LINKS
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Table of n, a(n) for n=1..63.
A. Hatcher, Stable Homotopy Groups of Spheres
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Exotic Sphere.
Wikipedia, Exotic sphere
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CROSSREFS
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Cf. A047680, A053381, A057617, A048648, A187595, A187717, A189995.
Sequence in context: A040778 A040780 A040781 * A040782 A040783 A057617
Adjacent sequences: A001673 A001674 A001675 * A001677 A001678 A001679
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KEYWORD
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nonn,hard,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Paul Muljadi, Mar 17 2011
Further terms from Jonathan Sondow, Jun 16 2011
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STATUS
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approved
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