cp's OEIS Frontend

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.

Previous Showing 11-13 of 13 results.

A131445 Numerators of n-th approximation of factorial (also called harmonic) expansion of Pi.

Original entry on oeis.org

3, 3, 3, 25, 47, 2261, 15833, 42223, 42223, 11400211, 1672031, 136802537, 2173640311, 2173640311, 342348348983, 5975534818613, 372475003693547, 21511925347007, 76431870757915873, 56199904969055789, 4223866541884824563
Offset: 1

Views

Author

Wolfdieter Lang, Aug 07 2007

Keywords

Comments

Denominators are given in A131446. Rationals in lowest terms.

Examples

			Rationals r(n): 3, 3, 3, 25/8, 47/15, 2261/720, 15833/5040, 42223/13440, 42223/13440, ...

Links

Formula

a(n) = numerator(r(n)), with r(n) = Sum_{k=1..n} b(k)/k! with b(k) = A075874(k) (factorial expansion of Pi).

A068448 Factorial expansion of log(Pi) = Sum_{n>0} a(n)/n! with a(n) as large as possible.

Original entry on oeis.org

1, 0, 0, 3, 2, 2, 1, 3, 4, 5, 8, 10, 11, 7, 13, 13, 3, 14, 11, 16, 6, 9, 3, 14, 0, 16, 22, 9, 8, 26, 5, 18, 6, 3, 13, 31, 4, 27, 25, 5, 12, 1, 17, 31, 2, 4, 16, 17, 39, 15, 15, 25, 52, 40, 50, 3, 27, 32, 54, 18, 55, 10, 29, 62, 38, 4, 17, 53, 13, 24, 22, 40, 23, 11, 74, 18, 74, 31, 8
Offset: 1

Views

Author

Benoit Cloitre, Mar 10 2002

Keywords

Comments

If a(n) is not required to be as large as possible, it isn't well defined: it can be decreased by any amount x without changing the value of the sum, if x*(n+1) is added to a(n+1), which in turn can be decreased by any arbitrary amount etc. - M. F. Hasler, Dec 04 2018

Links

Crossrefs

Cf. A053510 (decimal expansion).
Similar expansions: A068450 (sqrt(Pi)), A075874 (Pi), A007514 (a different variant for Pi).

Programs

  • Magma
    R:= RealField(); [Floor(Log(Pi(R)))] cat [Floor(Factorial(n)*Log(Pi(R))) - n*Floor(Factorial((n-1))*Log(Pi(R))) : n in [2..30]]; // G. C. Greubel, Mar 21 2018
  • Mathematica
    Table[If[n == 1, Floor[Log[Pi]], Floor[n!*Log[Pi]] - n*Floor[(n - 1)!*Log[Pi]]], {n,1,50}] (* G. C. Greubel, Mar 21 2018 *)
  • PARI
    for(n=1,30, print1(if(n==1, floor(log(Pi)), floor(n!*log(Pi)) - n*floor((n-1)!*log(Pi))), ", ")) \\ G. C. Greubel, Mar 21 2018
  • PARI
    A068448_vec(N=90,c=log(precision(Pi,N*log(N/2.4)\/2.3)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ N*log(N/2.4)\/2.3 ~ logint(N!,10) but uses much less memory when N is big. - M. F. Hasler, Nov 28 2018

Extensions

Name edited by M. F. Hasler, Dec 04 2018

A131446 Denominators of n-th approximation of factorial (also called harmonic) expansion of Pi.

Original entry on oeis.org

1, 1, 1, 8, 15, 720, 5040, 13440, 13440, 3628800, 532224, 43545600, 691891200, 691891200, 108972864000, 1902071808000, 118562476032000, 6847458508800, 24329020081766400, 17888985354240000, 1344498478202880000
Offset: 1

Views

Author

Wolfdieter Lang, Aug 07 2007

Keywords

Comments

Numerators are given in A131445.
For the rationals r(n) and some values see the W. Lang link under A131445.

Examples

			Rationals r(n): [3,3,3,25/8,47/15,2261/720,15833/5040,42223/13440,42223/13440,...].

Crossrefs

Cf. A000796, A131445.

Formula

a(n) = denominator(r(n)), with r(n):=sum(b(k)/k!,n=1..n) with b(k):=A075874(k) (factorial expansion of Pi).
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.