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%I #38 Oct 10 2019 04:37:46
%S 3,15,105,385,5005,85085,323323,7436429,30808063,955049953,
%T 35336848261,1448810778701,5663533044013,266186053068611,
%U 1085220062510491,64027983688118969,3905707004975257109,15393080549020130959,1092908718980429298089,79782336485571338760497
%N Numerators of the partial Euler product representation of Pi/4.
%C 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.
%H N. Elkies, <a href="http://www.math.harvard.edu/~elkies/M259.06/dirichlet.pdf">Introduction to Analytic Number Theory: Primes in Arithmetic Progression, Dirichlet Characters and L-Functions</a>
%H L. Euler, <a href="https://arxiv.org/abs/math/0506415">On the sums of series of reciprocals</a>, arXiv:math/0506415 [math.HO], 2005-2008.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Superparticular_ratio">Superparticular ratio</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wallis_product">Wallis Product</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gregory%27s_series">Gregory Series</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Madhava_series">Madhava Series</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Machin-like_formula">Machin-like Formula</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions">Inverse Trigonometric Functions</a>
%e a(3) = 105 = numerator((3/4) * (5/4) * (7/8)).
%o (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
%Y Cf. A003881 (Decimal expansion of Pi/4).
%Y Cf. A101455 (Dirichlet L-series of The Non-Principal Dirichlet Characters Mod 4).
%Y Cf. A323552 (Denominators of the Partial Euler Product Representation of Pi/4).
%Y Cf. A236436 (Denominators of the Product (1 + 1/p), where p is prime).
%Y Cf. A002144 (Primes of the form 4n+1; Pythagorean primes).
%Y Cf. A002145 (Primes of the form 4n+3).
%K nonn,frac
%O 1,1
%A _Anthony Hernandez_, Jan 16 2019
%E More terms from _Daniel Suteu_, Jan 22 2019
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