A242032 - OEIS

%I #33 Jan 05 2025 19:51:40

%S 2,2,7,31,127,73,1414477,8191,16931177,5749691557,91546277357,

%T 3324754717,1982765468311237,22076500342261,65053034220152267,

%U 925118910976041358111,16555640865486520478399,8089941578146657681,29167285342563717499865628061

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

%C 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)).

%C 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.

%C 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.

%H Helaman Rolfe Pratt Ferguson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-1/ferguson.pdf"> Bernoulli Numbers and Non-Standard Differentiable Structures on (4k-1)-Spheres</a>, The Fibonacci Quarterly, Vol. 11(1), 1973.

%H Friedrich Hirzebruch, <a href="http://dx.doi.org/10.1007/978-3-662-34515-3">Neue topologische Methoden in der algebraischen Geometrie</a>, 1956, Springer Berlin.

%H Friedrich Hirzebruch, <a href="http://www.numdam.org/item?id=SB_1966-1968__10__13_0">Singularities and exotic spheres</a> Séminaire Bourbaki, 10 (1966-1968), Exp. No. 314.

%H John Milnor, <a href="http://www.jstor.org/stable/2372998">Differentiable Structures on Spheres</a>, American Journal of Mathematics, Vol. 81, No. 4 (Oct., 1959), pp. 962-972.

%H John W. Milnor and Michel A. Kervaire, <a href="https://citeseerx.ist.psu.edu/pdf/dbced51695c22f06354f7d4bb8fc88f61511a059">Bernoulli numbers, homotopy groups, and a theorem of Rohlin</a>, J. A. Todd (ed.) , Proc. Internat. Congress Mathematicians (Edinburgh, 1958) , Cambridge Univ. Press (1960) pp. 454-458.

%H Dinesh S. Thakur, <a href="http://dx.doi.org/10.1090/S0002-9939-2012-11234-1">A note on numerators of Bernoulli numbers</a>, Proc. Amer. Math. Soc. 140 (2012), 3673-3676.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exotic_sphere">Exotic sphere</a>

%e a( 0) = 2

%e a( 1) = 2

%e a( 2) = 7

%e a( 3) = 31

%e a( 4) = 127

%e a( 5) = 73

%e a( 6) = 23 * 89 * 691

%e a( 7) = 8191

%e a( 8) = 31 * 151 * 3617

%e a( 9) = 43867 * 131071

%e a(10) = 283 * 617 * 524287

%e a(11) = 127 * 131 * 337 * 593

%e a(12) = 47 * 103 * 178481 * 2294797

%e a(13) = 31 * 601 * 1801 * 657931

%t h[x_] := Zeta[2x] (4^x-2);

%t 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];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 12 2019, from Sage code *)

%o (Sage)

%o def A242032(n):

%o     h = lambda x: zeta(2*x)*(4^x-2)

%o     M = Integer((h((n+1)//2)*h(n//2)/h(n)).denominator())

%o     k = 2*((n+1)//2)

%o     P = Primes()

%o     p = P.first()

%o     while p < k:

%o         while p.divides(M):

%o             M /= p

%o         p = P.next(p)

%o     return M

%o [A242032(n) for n in (0..30)]

%Y Cf. A036280, A246053, A240978.

%K nonn,changed

%O 0,1

%A _Peter Luschny_, Aug 12 2014