cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A191996 Numerators of partial products of a Hardy-Littlewood constant.

Original entry on oeis.org

2, 3, 45, 175, 693, 11011, 2807805, 302307005, 402243205, 714186915, 42803602439, 11086133031701, 5908908905896633, 1488200914442251997, 3041106216468949733, 16213234917387714257, 21611220383343195817, 77778782159652161745383, 67745319261057032880228593
Offset: 2

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Author

Wolfdieter Lang, Jun 21 2011

Keywords

Comments

The rational partial products are r(n)=a(n)/A191997(n), n>=1.
The limit r(n), n->infinity, approximately 1.3203236 = A114907, is the constant C(f_1,f_2) appearing in the Hardy-Littlewood conjecture (also called Bateman-Horn conjecture) for the integer polynomials f_1=x and f_2=x+2 (relevant for twin primes). See the Conrad reference Example 1, p. 134, also for the original references.
Essentially the same as A062270. - R. J. Mathar, Jun 23 2011

Examples

			The rationals r(n) (in lowest terms) are 2, 3/2, 45/32, 175/128, 693/512, 11011/8192,...

References

  • Keith Conrad, Hardy-Littlewood constants, pp. 133-154 in: Mathematical properties of sequences and other combinatorial structures, edts. Jong-Seon No et al., Kluwer, Boston/Dordrecht/London, 2003.

Links

Crossrefs

A191997, A191998/A191999.

Formula

a(n) = numerator(r(n)), with the rational r(n):=2*product(1-1/(p(j)-1)^2,j=2..n), with the primes p(j):=A000040(j).

A191998 Numerators of partial products of a Hardy-Littlewood constant.

Original entry on oeis.org

1, 3, 9, 21, 231, 847, 2541, 16093, 33649, 43263, 447051, 1043119, 13560547, 83300503, 170222767, 222599003, 13133341177, 774867129443, 4719645242971, 335094812250941
Offset: 1

Views

Author

Wolfdieter Lang, Jun 21 2011

Keywords

Comments

The rational partial products are r(n) = a(n)/A191999(n), n >= 1.
The limit r(n), n->infinity, approximately 1.3728134 is the constant C(f) appearing in the Hardy-Littlewood conjecture (also called Bateman-Horn conjecture) for the integer polynomial f=x^2+1. See the Conrad reference Example 2, p. 134, also for the original references.
Note that the nontrivial Dirichlet character modulo 4, called Chi_2(4;n) = A056594(n-1), n >= 1, appears as Chi_4(n) in this reference. The constant 0.6864067 given there is C(f)/2 (the degree of the function f has been divided).

Examples

			The rationals r(n) are: 1, 3/2, 9/8, 21/16, 231/160, 847/640, 2541/2048, ...

References

  • Keith Conrad, Hardy-Littlewood constants, pp. 133-154 in: Mathematical properties of sequences and other combinatorial structures, edts. Jong-Seon No et al., Kluwer, Boston/Dordrecht/London, 2003.

Links

Crossrefs

Cf. A191999, A191996/A191997.

Formula

a(n) = numerator(r(n)) with
r(n) := product(1-Chi_2(4;p(j))/(p(j)-1),j=1..n), n>=1, with the primes p(j)=A000040(j) and the nontrivial Dirichlet Character modulo 4, called here Chi_2(4;k) = A056594(k).

A191999 Denominators of partial products of a Hardy-Littlewood constant.

Original entry on oeis.org

1, 2, 8, 16, 160, 640, 2048, 12288, 24576, 32768, 327680, 786432, 10485760, 62914560, 125829120, 167772160, 9730785280, 583847116800, 3503082700800, 245215789056000
Offset: 1

Views

Author

Wolfdieter Lang, Jun 21 2011

Keywords

Comments

See A191998.

Examples

			The rationals r(n) are: 1, 3/2, 9/8, 21/16, 231/160, 847/640, 2541/2048, ...

Crossrefs

Cf. A191998, A191996/A191997.

Formula

a(n) = denominator(r(n)) with
r(n):=product(1-Chi_2(4;p(j))/(p(j)-1),j=1..n), n>=1, with the primes p(j)=A000040(j) and the nontrivial Dirichlet Character modulo 4, called here Chi_2(4;k) = A056594(k).
Showing 1-3 of 3 results.

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