A242032 A sequence related to lower bounds for the number of distinct differentiable structures on spheres of the form S^(4*k-1).

2, 2, 7, 31, 127, 73, 1414477, 8191, 16931177, 5749691557, 91546277357, 3324754717, 1982765468311237, 22076500342261, 65053034220152267, 925118910976041358111, 16555640865486520478399, 8089941578146657681, 29167285342563717499865628061

Offset: 0

Author

Peter Luschny, Aug 12 2014

Keywords

• nonn

Comments

Definition: Divide the prime factorization of an integer M into two parts: L(k, M) = product{p^v(M) the highest power of a prime p which divides M and p < k} and H(k, M) = M / L(k, M). Then a(n) = H(2*floor((n+1)/2), A246053(n)).

For some n the a(n) are lower bounds for the number of distinct differentiable structures on spheres. Compare theorem 2 of the Milnor 1959 paper which asserts that a(2) and a(4) through a(8) are lower bounds for the spheres S^(4*k-1) for k = 2 and 4,..,8.

Let b(n) = numerator(B(2*n)/(2*n)!*(4^n-2)*(-1)^(n-1)), B(n) Bernoulli number. Apart from n=0 and n=1 the first n such that a(n) != b(n) is n = 1437. Thus in the range [2..1436] the a(n) are the numerators in the Taylor series for x*cosec(x), A036280.

Examples

			a( 0) = 2
a( 1) = 2
a( 2) = 7
a( 3) = 31
a( 4) = 127
a( 5) = 73
a( 6) = 23 * 89 * 691
a( 7) = 8191
a( 8) = 31 * 151 * 3617
a( 9) = 43867 * 131071
a(10) = 283 * 617 * 524287
a(11) = 127 * 131 * 337 * 593
a(12) = 47 * 103 * 178481 * 2294797
a(13) = 31 * 601 * 1801 * 657931
		

Links

• Helaman Rolfe Pratt Ferguson, Bernoulli Numbers and Non-Standard Differentiable Structures on (4k-1)-Spheres, The Fibonacci Quarterly, Vol. 11(1), 1973.
• Friedrich Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, 1956, Springer Berlin.
• Friedrich Hirzebruch, Singularities and exotic spheres Séminaire Bourbaki, 10 (1966-1968), Exp. No. 314.
• John Milnor, Differentiable Structures on Spheres, American Journal of Mathematics, Vol. 81, No. 4 (Oct., 1959), pp. 962-972.
• John W. Milnor and Michel A. Kervaire, Bernoulli numbers, homotopy groups, and a theorem of Rohlin, J. A. Todd (ed.) , Proc. Internat. Congress Mathematicians (Edinburgh, 1958) , Cambridge Univ. Press (1960) pp. 454-458.
• Dinesh S. Thakur, A note on numerators of Bernoulli numbers, Proc. Amer. Math. Soc. 140 (2012), 3673-3676.
• Wikipedia, Exotic sphere

Crossrefs

Cf. A036280, A246053, A240978.

Programs

Mathematica

h[x_] := Zeta[2x] (4^x-2);
a[n_] := Module[{M, k, p}, M = Denominator[h[Quotient[n+1, 2]] h[Quotient[ n, 2]]/h[n]]; k = 2 Quotient[n+1, 2]; p = 2; While[p < k, While[ Divisible[M, p], M = M/p]; p = NextPrime[p]]; M];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 12 2019, from Sage code *)

Sage

def A242032(n):
    h = lambda x: zeta(2*x)*(4^x-2)
    M = Integer((h((n+1)//2)*h(n//2)/h(n)).denominator())
    k = 2*((n+1)//2)
    P = Primes()
    p = P.first()
    while p < k:
        while p.divides(M):
            M /= p
        p = P.next(p)
    return M
[A242032(n) for n in (0..30)]