A246051 -id:A246051 - 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)).

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)]

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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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.

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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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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).

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Sage

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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.

Views

Author

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Comments

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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

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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).

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).