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) * (17/16) * (19/20) * ... with each term in the product being the ratio of a prime number to its nearest multiple of 4. The sequence consists of the denominators 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) = 128 = denominator((3/4) * (5/4) * (7/8)).
PROG
(PARI) a(n) = denominator(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
KEYWORD
nonn,frac
AUTHOR
Anthony Hernandez, Jan 17 2019
EXTENSIONS
More terms from Daniel Suteu, Jan 22 2019
STATUS
approved