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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A323551 Numerators of the partial Euler product representation of Pi/4.

Original entry on oeis.org

3, 15, 105, 385, 5005, 85085, 323323, 7436429, 30808063, 955049953, 35336848261, 1448810778701, 5663533044013, 266186053068611, 1085220062510491, 64027983688118969, 3905707004975257109, 15393080549020130959, 1092908718980429298089, 79782336485571338760497
Offset: 1

Views

Author

Anthony Hernandez, Jan 16 2019

Keywords

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.

Examples

			a(3) = 105 = numerator((3/4) * (5/4) * (7/8)).

Links

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).

Programs

  • 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

Extensions

More terms from Daniel Suteu, Jan 22 2019
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