

A323551


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


1



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

1,1


COMMENTS

The Euler product representation follows from the classical Leibniz series representation of Pi/4 interpreted as a Dirichlet Lseries using the unique nonprincipal 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 LFunctions
L. Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 20052008.
Wikipedia, Superparticular ratio
Wikipedia, Wallis Product
Wikipedia, Gregory Series
Wikipedia, Madhava Series
Wikipedia, Machinlike 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/(p1), p/(p+1)))); \\ Daniel Suteu, Jan 22 2019


CROSSREFS

Cf. A003881 (Decimal expansion of Pi/4).
Cf. A101455 (Dirichlet Lseries of The NonPrincipal 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



