A246053
The denominator of h(n-k)*h(k)/h(n) where h(x) = zeta(2*x)*(4^x-2) and k = floor(n/2).
Original entry on oeis.org
2, 2, 7, 62, 381, 365, 1414477, 573370, 118518239, 5749691557, 274638832071, 768018339627, 29741482024668555, 860983513348179, 65053034220152267, 1850237821952082716222, 16555640865486520478399, 962703047799452264039, 29167285342563717499865628061
Offset: 0
a( 0) = 2
a( 1) = 2
a( 2) = 7
a( 3) = 2 * 31
a( 4) = 3 * 127
a( 5) = 5 * 73
a( 6) = 23 * 89 * 691
a( 7) = 2 * 5 * 7 * 8191
a( 8) = 7 * 31 * 151 * 3617
a( 9) = 43867 * 131071
a(10) = 3 * 283 * 617 * 524287
a(11) = 3 * 7 * 11 * 127 * 131 * 337 * 593
a(12) = 3 * 5 * 47 * 103 * 178481 * 2294797
a(13) = 3 * 13 * 31 * 601 * 1801 * 657931
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.
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
2
2, 2
2, 7, 2
2, 62, 62, 2
2, 381, 381, 381, 2
2, 5110, 365, 365, 5110, 2
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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);
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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)]
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
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
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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)]
Showing 1-3 of 3 results.
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