A189995 The order b_{4n-1} of the cyclic group S_{4n-1}^{bp} of oriented diffeomorphism classes of smooth homotopy (4n-1)-spheres that bound parallelizable manifolds, for n > 1.

28, 992, 8128, 261632, 1448424448, 67100672, 1941802827776, 753623571759104, 23998307331473408, 341653284209033216, 8316321134799694594048, 740764429532373450752, 30559446583872811817762816, 496669433444154134078771167232, 17776484020396435145889494859776, 11188223110510348416175908585472

Offset: 2

Author

Jonathan Sondow, Jun 15 2011

Keywords

• nonn

Comments

For a(n), Milnor 2011 Theorem 5 gives the formula

2^(2*n-2)*(2^(2*n-1)-1)*numerator(4*bernoulli(n)/n)

where bernoulli(n) = abs(Bernoulli(2*n)).

See A001676 for additional comments, references, and links.

Examples

			a(2) = 2^2 * (2^3 - 1) * abs(numerator(4 * Bernoulli(4)/2)) = 4 * 7 * abs(numerator(2 * (-1/30))) = 28
		

References

• J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.

Links

• G. C. Greubel, Table of n, a(n) for n = 2..235
• John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809 (see Theorem 5 and Table 3).
• John W. Milnor, Spheres, Abel Prize lecture (video), 2011.

Crossrefs

Cf. A001676, A047680, A057617, A187595, A228689.

Programs

Magma

[2^(2*n-2)*(2^(2*n-1)-1)*Abs(Numerator(4*Bernoulli(2*n)/n)): n in [2..30]]; // G. C. Greubel, Jan 11 2018

Mathematica

Table[2^(2*n-2)*(2^(2*n-1)-1)*Abs[Numerator[4*BernoulliB[2*n]/n]],{n,2,17}]

Formula

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

a(n) = A187595(4*n-1) for n > 1.