A112407 -id:A112407 - cp's OEIS frontend

A090986 Decimal expansion of Pi/sinh(Pi).

2, 7, 2, 0, 2, 9, 0, 5, 4, 9, 8, 2, 1, 3, 3, 1, 6, 2, 9, 5, 0, 2, 3, 6, 5, 8, 3, 6, 7, 2, 0, 3, 7, 5, 5, 5, 8, 4, 0, 7, 1, 8, 3, 6, 3, 4, 6, 0, 3, 1, 5, 9, 4, 9, 5, 0, 6, 8, 9, 6, 7, 8, 3, 8, 5, 6, 2, 4, 6, 1, 9, 1, 3, 6, 9, 4, 8, 7, 8, 8, 8, 1, 9, 1, 1, 5, 3, 1, 1, 7, 2, 1, 0, 6, 9, 3, 7, 6, 4, 4, 8, 6, 1, 0

Offset: 0

Benoit Cloitre, Feb 28 2004

Or, decimal expansion of Pi * csch(Pi).

			0.272029054982133162950236583672...

Jonathan M. Borwein, David H. Bailey, and Roland Girgensohn, "Two Products", Section 1.2 in Experimentation in Mathematics: Computational Paths to Discovery, Natick, MA: A. K. Peters, 2004, pp. 4-7.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A112407, A144663, A144664, A144665, A144666, A144667, A144668, A144669, A212879.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sinh(Pi(R)); // G. C. Greubel, Feb 02 2019

Re[N[Gamma[1+I]*Gamma[1-I], 104]] (* Vaclav Kotesovec, Dec 09 2015 *)
RealDigits[Pi/Sinh[Pi],10,120][[1]] (* Harvey P. Dale, May 16 2019 *)

default(realprecision, 100);  Pi/sinh(Pi) \\ G. C. Greubel, Feb 02 2019

numerical_approx(pi/sinh(pi), digits=100) # G. C. Greubel, Feb 02 2019

Equals Pi/sinh(Pi) = Product_{k>=1} k^2/(k^2+1).
Equals Pi * csch(Pi) = Product_{n >= 2} (n^2 - 1)/(n^2 + 1). - Jonathan Vos Post, Dec 07 2005
Equals Gamma(1+i)*Gamma(1-i), where i is the imaginary unit. - Vaclav Kotesovec, Dec 10 2015
Equals (1)(-i)*(1)_i where (n)_k denotes the rising factorial. - _Peter Luschny, May 06 2022
Equals 1 - 2*Sum_{n >= 1} (-1)^(n+1)/(n^2 + 1). - Peter Bala, Jan 01 2023
Equals A212879^2. - Amiram Eldar, Oct 25 2024

A155799 Decimal expansion of the product_{q=3-almost-primes} (q^2-1)/(q^2+1).

Original entry on oeis.org

9, 2, 5, 8, 5, 7, 2, 7, 4, 7, 1, 2, 8, 9, 3, 1, 2, 7, 9, 9, 8, 8, 8, 2, 1, 3, 8, 2, 0, 7, 1, 5, 8, 4, 1, 5, 2, 7, 8, 4, 5, 0, 2, 1, 8, 1, 9, 1, 9, 6, 6, 0, 2, 1, 5, 3, 2, 7, 6, 5, 6, 6, 2, 0, 2, 9, 5, 6, 7, 4, 4, 6, 8, 1, 0, 7, 1, 2, 4, 7, 5, 7, 0, 3, 9, 6, 4, 4, 8, 6, 6, 8, 9

Offset: 0

R. J. Mathar, Jan 27 2009

The 3-almost-prime analog of A112407. Its logarithm has been computed from -2*sum_{l=1..infinity} P_3(2*(2l-1))/(2l-1) where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

			0.92585727... = 63/65*143/145*323/325*399/401*364/365*...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT].
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], third line Table 1. [From _R. J. Mathar_, Mar 28 2009]

Cf. A112407.

product_{n=1..infinity} (A014612(n)^2-1)/(A014612(n)^2+1).

A385809 Decimal expansion of the Product_{p prime} (p^3-1)/(p^3+1).

Original entry on oeis.org

7, 0, 4, 0, 7, 2, 4, 8, 7, 3, 2, 0, 7, 8, 4, 4, 7, 8, 2, 9, 6, 2, 9, 8, 1, 9, 9, 9, 7, 8, 6, 2, 4, 4, 5, 8, 0, 9, 2, 5, 8, 3, 7, 8, 1, 1, 1, 9, 9, 8, 8, 2, 9, 3, 2, 4, 2, 8, 8, 4, 6, 9, 1, 1, 8, 9, 5, 3, 7, 1, 8, 6, 8, 7, 7, 9, 9, 1, 6, 3, 3, 0, 9, 4, 9, 4, 9, 0, 7, 4, 2, 0, 3, 0, 8, 2, 8, 1, 3, 9, 7, 5, 4, 1, 9, 9, 5, 5, 0, 8

Offset: 0

Artur Jasinski, Aug 01 2025

Product_{p prime} (p^(2*n)-1)/(p^(2*n)+1) are rational numbers A114362(n)/A114363(n) = zeta(4*n)/zeta(2*n)^2.

Product_{p prime} (p^(2*n+1)-1)/(p^(2*n+1)+1) = zeta(2*(2*n+1))/zeta(2*n+1)^2.

			0.70407248732078447829629819997862445809258378...

Richard Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011, Table 1 p. 2.

Cf. A002117, A013664, A112407, A114362, A114363, A376742.

RealDigits[Zeta[6]/Zeta[3]^2,10,105][[1]]

PARI
```
prodeulerrat((p^3-1)/(p^3+1))
```

Equals zeta(6)/zeta(3)^2.

Equals 1 / A376742. - Amiram Eldar, Aug 01 2025

a(109) corrected by Georg Fischer, Aug 31 2025

cp's OEIS Frontend

A090986 Decimal expansion of Pi/sinh(Pi).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A155799 Decimal expansion of the product_{q=3-almost-primes} (q^2-1)/(q^2+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A385809 Decimal expansion of the Product_{p prime} (p^3-1)/(p^3+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions