A323551 Numerators of the partial Euler product representation of Pi/4.

3, 15, 105, 385, 5005, 85085, 323323, 7436429, 30808063, 955049953, 35336848261, 1448810778701, 5663533044013, 266186053068611, 1085220062510491, 64027983688118969, 3905707004975257109, 15393080549020130959, 1092908718980429298089, 79782336485571338760497

Offset

1,1

Comments

The Euler product representation follows from the classical Leibniz series representation of Pi/4 interpreted as a Dirichlet L-series using the unique non-principal Dirichlet characters modulo 4, whose (infinite) Euler product representation can be written as (3/4) * (5/4) * (7/8) * (11/12) * (13/12) * ..., with each term in the product being the ratio of a prime number to its nearest multiple of 4. The sequence consists of the numerators of the partial products.

Links

Table of n, a(n) for n=1..20.

N. Elkies, Introduction to Analytic Number Theory: Primes in Arithmetic Progression, Dirichlet Characters and L-Functions

L. Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.

Wikipedia, Superparticular ratio

Wikipedia, Wallis Product

Wikipedia, Gregory Series

Wikipedia, Madhava Series

Wikipedia, Machin-like Formula

Wikipedia, Inverse Trigonometric Functions

Example

a(3) = 105 = numerator((3/4) * (5/4) * (7/8)).

Prog

(PARI) a(n) = numerator(prod(k=2, n+1, my(p=prime(k)); if(p%4==1, p/(p-1), p/(p+1)))); \\ Daniel Suteu, Jan 22 2019

Crossrefs

Cf. A003881 (Decimal expansion of Pi/4).

Cf. A101455 (Dirichlet L-series of The Non-Principal Dirichlet Characters Mod 4).

Cf. A323552 (Denominators of the Partial Euler Product Representation of Pi/4).

Cf. A236436 (Denominators of the Product (1 + 1/p), where p is prime).

Cf. A002144 (Primes of the form 4n+1; Pythagorean primes).

Cf. A002145 (Primes of the form 4n+3).

Sequence in context: A220747 A088989 A001801 * A267840 A067546 A015682

Adjacent sequences: A323548 A323549 A323550 * A323552 A323553 A323554

Keyword

nonn,frac

Author

Anthony Hernandez, Jan 16 2019

Extensions

More terms from Daniel Suteu, Jan 22 2019

Status

approved