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
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).
KEYWORD
nonn,frac
AUTHOR
Anthony Hernandez, Jan 16 2019
EXTENSIONS
More terms from Daniel Suteu, Jan 22 2019
STATUS
approved