A323552 Denominators of the partial Euler product representation of Pi/4.

4, 16, 128, 512, 6144, 98304, 393216, 9437184, 37748736, 1207959552, 43486543872, 1739461754880, 6957847019520, 333976656936960, 1335906627747840, 80154397664870400, 4809263859892224000, 19237055439568896000, 1385067991648960512000, 99724895398725156864000

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

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) = 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

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

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

Cf. A323551 (Numerators of the Partial Euler Product Representation of Pi/4).

Sequence in context: A322236 A318152 A358083 * A363443 A349264 A061129

Adjacent sequences: A323549 A323550 A323551 * A323553 A323554 A323555

Keyword

nonn,frac

Author

Anthony Hernandez, Jan 17 2019

Extensions

More terms from Daniel Suteu, Jan 22 2019

Status

approved