A189995 -id:A189995 - cp's OEIS frontend

A001676 Number of h-cobordism classes of smooth homotopy n-spheres.

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, 1, 8, 4, 1986677733776616536315084668928, 4, 1, 24, 284423744326342962334231917756416

Offset: 1

N. J. A. Sloane

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 Poincaré 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.
The sequence is essentially given in the rightmost column of tables 1 and 2 in Isaksen, Wang & Xu (2020). It corrects some errors in earlier work. - Andrey Zabolotskiy, Nov 27 2022

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 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.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..83 using data from Isaksen, Wang & Xu (2023).
Tom Copeland, The Kervaire-Milnor formula
Brady Haran and Ciprian Manolescu, The Puzzling Fourth Dimension (and exotic shapes), Numberphile video (2022).
Kevin Hartnett, An Old Conjecture Falls, Making Spheres a Lot More Complicated, Quanta Magazine, Aug 22 2023.
A. Hatcher, Stable Homotopy Groups of Spheres
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres, PNAS, 117 (2020), 24757-24763.
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres: from dimension 0 to 90, Publications mathématiques de l'IHÉS, 137 (2023), 107-243.
M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I, Ann. of Math. (2) 77 1963 504-537.
S. S. Khare, On Abel Prize 2011 to John Willard Milnor, Math. Student, 82 (2013), 247-279. [dead link]
Alexander Kupers, Lectures on diffeomorphism groups of manifolds, Version Apr 28 2018.
J. W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. 64 (1956), 399-405.
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.
G. D. Rizell, J. D. Evans, Exotic spheres and the topology of symplectomorphism groups, J. Topol. 8 (2015) 586-602
Anthony Saint-Criq, What is so exotic about dimension four?, Univ. de Toulouse (France 2022).
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

Cf. A053381, A057617, A048648, A187595, A187717, A189995, A191783, A228689, A228690, A228691, A228692.

More terms from Paul Muljadi, Mar 17 2011
Further terms from Jonathan Sondow, Jun 16 2011
The terms a(56), a(57), a(63) corrected by Andrey Zabolotskiy, Nov 27 2022

A053381 Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3

Offset: 0

Warren D. Smith, Jan 06 2000

The corresponding terms for a 2n-sphere are all 0 ("you can't comb the hair on a billiard ball"). The "3" and "7" come from the quaternions and octonions.
b(n) = a(n-1): b(2^e) = ((e+1) idiv 4) + 2^((e+1) mod 4) - 1, b(p^e) = 1, p>2. - Christian G. Bower, May 18 2005
a(n-1) is multiplicative. - Christian G. Bower, Jun 03 2005

T. D. Noe, Table of n, a(n) for n = 0..10000
J. Frank Adams, Vector fields on spheres, Topology, 1 (1962), 63-65.
J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41.
J. Frank Adams, Vector fields on spheres, Annals of Math. 75 (1962) 603-632.
A. Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
M. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA 44 (1958) 280-283.
J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
J. Radon, Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.

For another version see A003484. Cf. A189995, A001676.

Cf. A047530, A220466.

int MaxLinInd(int n){ /* Returns max # linearly indep smooth nowhere zero * vector fields on S^{n-1}, n=1,2,... */ int b,c,d,rho; b = 0; while((n & 1)==0){ n /= 2; b++; } c = b & 3; d = (b - c)/4; rho = (1 << c) + 8*d; return( rho - 1); }

int MaxLinInd(int n) { int b = _builtin_ctz(n); return (1<<b%4) + b/4*8 - 1; } /* _Jeremy Tan, Apr 09 2021 */

with(numtheory): for n from 1 to 601 by 2 do c := irem(ifactors(n+1)[2,1,2],4): d := iquo(ifactors(n+1)[2,1,2],4): printf(`%d,`, 2^c+8*d-1) od:
nmax:=101: A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do A053381((2*n+1)*2^p-1) := A047530(p+1): od: od: seq(A053381(n), n=0..nmax); # Johannes W. Meijer, Jun 07 2011, revised Jan 29 2013

a[n_] := Module[{b, c, d, rho, n0}, n0 = 2*n; b = 0; While[BitAnd[n0, 1] == 0, n0 /= 2; b++]; c = BitAnd[b, 3]; d = (b - c)/4; rho = 2^c + 8*d; Return[rho - 1]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, May 16 2013, translated from C *)

Let f(n) be the number of linearly independent smooth nowhere-zero vector fields on an n-sphere. Then f(n) = 2^c + 8d - 1 where n+1 = (2a+1) 2^b and b = c+4d and 0 <= c <= 3. f(n) = 0 if n is even.
a((2*n+1)*2^p-1) = A047530(p+1), p >= 0 and n >= 0. a(2*n) = 1, n >= 0, and a(2^p-1) = A047530(p+1), p >= 0. - Johannes W. Meijer, Jun 07 2011
a(n) = A209675(n+1) - 1. - Reinhard Zumkeller, Mar 11 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 10/3. - Amiram Eldar, Nov 29 2022

More terms from James Sellers, Jun 01 2000

A228689 a(n) = A001676(4*n + 3).

Original entry on oeis.org

1, 28, 992, 16256, 523264, 69524373504, 67100672, 7767211311104, 3014494287036416, 2303837503821447168, 341653284209033216, 798366828940770681028608, 11852230872517975212032, 91678339751618435453288448, 1986677733776616536315084668928, 284423744326342962334231917756416

Offset: 0

N. J. A. Sloane, Aug 30 2013

nonn

V. P. Snaith, A history of the Arf-Kervaire invariant problem, Notices Amer. Math. Soc., 60 (No. 8, 2013), 1040-1047.

Cf. A001676, A228690, A228691, A228692, A189995.

Term a(15) corrected by Andrey Zabolotskiy, Nov 27 2022

A047680 a(n) = 2^(2n-2) (2^(2n-1)-1) numerator(|Bernoulli(4*n)|/n).

Original entry on oeis.org

1, 28, 342736, 29398976, 22841912576, 495449389339648, 113828913905979392, 4138783908783021080576, 226044736813811127599528869888, 35882867158508587149623987273728, 5564179808875681941887049052192768, 197363311803624016534960016288517315362816

Offset: 1

N. J. A. Sloane

nonn

Index entries for sequences related to Bernoulli numbers

Cf. A189995 (similar formula).

(-1)*2^(2*n-2)*(2^(2*n-1)-1)*numer(bernoulli(4*n)/n);

a(n) = 2^(2*n-2) * (2^(2*n-1)-1) * numerator(abs(bernfrac(4*n))/n); \\ Michel Marcus, Nov 30 2022

Edited by Andrey Zabolotskiy, Nov 30 2022

cp's OEIS Frontend

A001676 Number of h-cobordism classes of smooth homotopy n-spheres.

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A053381 Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.

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A228689 a(n) = A001676(4*n + 3).

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A047680 a(n) = 2^(2*n-2) * (2^(2*n-1)-1) * numerator(|Bernoulli(4*n)|/n).

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A047680 a(n) = 2^(2n-2) (2^(2n-1)-1) numerator(|Bernoulli(4*n)|/n).