A246052 -id:A246052 - cp's OEIS frontend

A246053 The denominator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2) and k = floor(n/2).

2, 2, 7, 62, 381, 365, 1414477, 573370, 118518239, 5749691557, 274638832071, 768018339627, 29741482024668555, 860983513348179, 65053034220152267, 1850237821952082716222, 16555640865486520478399, 962703047799452264039, 29167285342563717499865628061

Offset: 0

Peter Luschny, Aug 12 2014

There are terms that are not squarefree. For example, a(113) is divisible by 631^2 and a(114) is divisible by 103^2. Most terms appear to be divisible by numerator(bernoulli(2*n)/factorial(2*n)) but not all. The first two exceptions are a(1437) and a(23766). - Hans Havermann, Aug 16 2014

			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

Hans Havermann, Table of n, a(n) for n = 0..200
Hans Havermann, Factorization table of n, a(n) for n = 0..150
Dinesh S. Thakur, A note on numerators of Bernoulli numbers, Proc. Amer. Math. Soc. 140 (2012), 3673-3676.

Cf. A242035 (numerator), A240978 (largest prime divisor), A242050, A246051, A246052.

h = lambda x: zeta(2*x)*(4^x-2)
A246053 = lambda n: Integer((h((n+1)//2)*h(n//2)/h(n)).denominator())
[A246053(n) for n in range(19)]

a(n) = A246052(n, floor(n/2)).

A240978 The largest prime divisor of A246053(n).

Original entry on oeis.org

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

A246051 Triangle read by rows: numerator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2), 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 49, 49, 1, 1, 310, 343, 310, 1, 1, 4191, 341, 341, 4191, 1, 1, 1162525, 2669667, 1374230, 2669667, 1162525, 1, 1, 1414477, 46501, 562991, 562991, 46501, 1414477, 1, 1, 13924700, 48092218, 1613300, 117628797, 1613300, 48092218, 13924700, 1

Offset: 0

Peter Luschny, Aug 11 2014

			Triangle starts:
             1
            1, 1
          1, 5, 1
        1, 49, 49, 1
    1, 310, 343, 310, 1
1, 4191, 341, 341, 4191, 1

Cf. A246052 (denominators).

h := x -> Zeta(2*x)*(4^x-2);
A246051 := (n, k) -> h(n-k)*h(k)/h(n);
seq(print(seq(numer(A246051(n, k)), k=0..n)), n=0..8);

h = lambda n: zeta(2*n)*(4^n-2)
A246051 = lambda n, k: h(n-k)*h(k)/h(n)
for n in range(8): [A246051(n, k).numerator() 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)]

cp's OEIS Frontend

A246053 The denominator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2) and k = floor(n/2).

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A240978 The largest prime divisor of A246053(n).

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A246051 Triangle read by rows: numerator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(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(2x)*(4^x-2) and k = floor(n/2).

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Author

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Sage

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Formula

A240978 The largest prime divisor of A246053(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Sage

Formula

A246051 Triangle read by rows: numerator of h(n-k)*h(k)/h(n) where h(x) = zeta(2*x)*(4^x-2), 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Sage

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).

Views

Author

Keywords

Crossrefs

Programs

Sage

A246053 The denominator of h(n-k)h(k)/h(n) where h(x) = zeta(2x)*(4^x-2) and k = floor(n/2).

A246051 Triangle read by rows: numerator 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).