A246053 -id:A246053 - cp's OEIS frontend

A240978 The largest prime divisor of A246053(n).

2, 2, 7, 31, 127, 73, 691, 8191, 3617, 131071, 524287, 593, 2294797, 657931, 362903, 1001259881, 2147483647, 151628697551, 26315271553053477373, 154210205991661, 1897170067619, 1520097643918070802691, 1798482437, 67568238839737, 153289748932447906241

Offset: 0

Peter Luschny, Aug 12 2014

nonn

According to theorem 2 of the Milnor paper a(2) and a(4) through a(8) are lower bounds for the number of distinct differentiable structures on spheres S^(4*k-1) for k = 2 and 4,..,8. Better bounds are given in A242032.

John Milnor, Differentiable Structures on Spheres, American Journal of Mathematics, Vol. 81, No. 4 (Oct., 1959), pp. 962-972. [See p. 971]

Cf. A246053, A246052, A242032.

h = lambda x: zeta(2*x)*(4^x-2)
A246053 = lambda n: Integer((h((n+1)//2)*h(n//2)/h(n)).denominator())
A240978 = lambda n: max(prime_divisors(A246053(n)))
[A240978(n) for n in range(25)]

a(n) = A006530(A246053(n)). - Michel Marcus, Aug 18 2014

A242050 Denominator of A242035(n) / A246053(floor(n/2)).

Original entry on oeis.org

2, 2, 2, 2, 1, 7, 1, 2, 1, 1, 1, 5, 1, 1, 7, 10, 1, 7, 1, 1, 1, 1, 3, 7, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 62, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Aug 17 2014

nonn

Let h(x) = zeta(2*x)*(4^x-2) and r(n) = h([(n+1)/2])*h([n/2])/ h(n). Then the denominator of r([n/2]) almost divides the numerator of r(n), where 'a almost divides b' means that a divides b or the denominator of the quotient b/a (reduced to lowest terms) is very small compared to the size of b.

			a(34) = 1 because A246053(17) divides A242035(34).
a(35) = 7 because A242035(35)/A246053(17) = 8416843291197378595121525029762095062482279733/7.

Peter Luschny, Table of n, a(n) for n = 0..10000

Cf. A242035, A246053.

h = lambda x: zeta(2*x)*(4^x-2)
r = lambda n: h((n+1)//2)*h(n//2)/h(n)
A242050 = lambda n: (r(n).numerator()/r(n//2).denominator()).denominator()
[A242050(n) for n in range(1000)]

A246052 Triangle read by rows: denominator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2), 0<=k<=n.

Original entry on oeis.org

2, 2, 2, 2, 7, 2, 2, 62, 62, 2, 2, 381, 381, 381, 2, 2, 5110, 365, 365, 5110, 2, 2, 1414477, 2828954, 1414477, 2828954, 1414477, 2, 2, 1720110, 49146, 573370, 573370, 49146, 1720110, 2, 2, 16931177, 50793531, 1638501, 118518239, 1638501, 50793531, 16931177, 2

Offset: 0

Peter Luschny, Aug 11 2014

Conjecture: A240978(n) divides T(n,k) for k in (1..n-1) and n>=2.

			             2
            2, 2
          2, 7, 2
        2, 62, 62, 2
    2, 381, 381, 381, 2
2, 5110, 365, 365, 5110, 2

Cf. A246051 (numerators), A240978, A246053.

h := x -> Zeta(2*x)*(4^x-2);
A246052 := (n, k) -> denom(h(n-k)*h(k)/h(n));
seq(print(seq(A246052(n, k), k=0..n)), n=0..8);

h = lambda n: zeta(2*n)*(4^n-2)
A246052 = lambda n, k: (h(n-k)*h(k)/h(n)).denominator()
for n in range(8): [A246052(n, k) for k in (0..n)]

A242035 The numerator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2) and k = floor(n/2).

Original entry on oeis.org

1, 1, 5, 49, 343, 341, 1374230, 562991, 117628797, 5722552563, 274111769750, 767094923209, 29727071936873882, 860722536439030, 65045120396044500, 1850097086237495825037, 16555136396811464938269, 962684710425111932621, 29167062964422333027973288250

Offset: 0

Peter Luschny, Aug 12 2014

Cf. A246053 (denominator), A242050, A246051, A246052.

h = lambda x: zeta(2*x)*(4^x-2)
A242035 = lambda n: Integer((h((n+1)//2)*h(n//2)/h(n)).numerator())
[A242035(n) for n in range(19)]

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

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Aug 12 2014

nonn

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.

			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

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

Cf. A036280, A246053, A240978.

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 *)

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)]

cp's OEIS Frontend

A240978 The largest prime divisor of A246053(n).

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A242050 Denominator of A242035(n) / A246053(floor(n/2)).

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A246052 Triangle read by rows: denominator of h(n-k)*h(k)/h(n) where h(x) = zeta(2*x)*(4^x-2), 0<=k<=n.

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A242035 The numerator of h(n-k)*h(k)/h(n) where h(x) = zeta(2*x)*(4^x-2) and k = floor(n/2).

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

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Mathematica

Sage

A246052 Triangle read by rows: denominator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2), 0<=k<=n.

A242035 The numerator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2) and k = floor(n/2).