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.

Showing 1-4 of 4 results.

A067882 Factorial expansion of log(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

0, 1, 1, 0, 3, 1, 0, 3, 6, 2, 5, 4, 6, 11, 4, 11, 5, 12, 3, 5, 13, 2, 22, 6, 22, 13, 20, 7, 1, 0, 1, 20, 2, 6, 4, 1, 18, 14, 35, 2, 11, 31, 16, 19, 42, 36, 41, 0, 14, 31, 25, 43, 4, 13, 34, 53, 50, 57, 2, 30, 12, 25, 45, 24, 2, 39, 57, 51, 30, 41, 65, 15, 9, 55, 23, 4, 35, 18, 77, 43
Offset: 1

Views

Author

Benoit Cloitre, Mar 10 2002

Keywords

Examples

			log(2) = 0 + 1/2! + 1/3! + 0/4! + 3/5! + 1/6! + 0/7! + 3/8! + 6/9! + ...

Links

Crossrefs

Cf. A002162 (decimal expansion), A016730 (continued fraction).
Cf. A322334 (log(3)), A322333 (log(5)), A068460 (log(7)), A068461 (log(11)).

Programs

  • Magma
    SetDefaultRealField(RealField(250)); [Floor(Log(2))] cat [Floor(Factorial(n)*Log(2)) - n*Floor(Factorial((n-1))*Log(2)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018
  • Mathematica
    With[{b = Log[2]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)
  • PARI
    default(realprecision, 250); b = log(2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018
  • Sage
    def A067882(n):
    if (n==1): return floor(log(2))
    else: return expand(floor(factorial(n)*log(2)) - n*floor(factorial(n-1)*log(2)))
[A067882(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Formula

a(n) = floor(n!*log(2)) - n*floor((n-1)!*log(2)).

A068460 Factorial expansion of log(7) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 0, 3, 0, 8, 8, 2, 11, 1, 5, 11, 1, 7, 1, 11, 16, 12, 12, 13, 5, 4, 26, 19, 12, 20, 0, 18, 14, 22, 6, 29, 0, 27, 16, 23, 23, 23, 34, 27, 4, 27, 18, 0, 10, 27, 42, 24, 9, 16, 6, 52, 2, 38, 44, 30, 51, 61, 7, 19, 0, 45, 18, 51, 43, 54, 7, 15, 69, 44, 51, 9, 74, 5, 69
Offset: 1

Views

Author

Benoit Cloitre, Mar 10 2002

Keywords

Examples

			log(7) = 1 + 1/2! + 2/3! + 2/4! + 3/5! + 3/6! + 0/7! + 3/8! + 0/9! + ...

Links

Crossrefs

Cf. A016630 (decimal expansion), A016735 (continued fraction).
Cf. A067882 (log(2)), A322334 (log(3)), A322333 (log(5)), A068461 (log(11)).

Programs

  • Magma
    SetDefaultRealField(RealField(250));  [Floor(Log(7))] cat [Floor(Factorial(n)*Log(7)) - n*Floor(Factorial((n-1))*Log(7)) : n in [2..80]]; // G. C. Greubel, Dec 05 2018
  • Maple
    Digits:=200: a:=n->`if`(n=1,floor(log(7)),floor(factorial(n)*log(7))-n*floor(factorial(n-1)*log(7))); seq[120](a(n),n=1..80); # Muniru A Asiru, Dec 06 2018
  • Mathematica
    With[{b = Log[7]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 05 2018 *)
  • PARI
    vector(30,n,if(n>1,t=t%1*n,t=log(7))\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018
  • PARI
    default(realprecision, 250); b = log(7); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Dec 05 2018
  • Sage
    def a(n):
    if (n==1): return floor(log(7))
    else: return expand(floor(factorial(n)*log(7)) - n*floor(factorial(n-1)*log(7)))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 05 2018

Extensions

Name edited and keywords cons,easy removed by M. F. Hasler, Nov 25 2018

A068461 Factorial, or factoradic, expansion of log(11) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

Original entry on oeis.org

2, 0, 2, 1, 2, 4, 3, 3, 1, 2, 4, 0, 3, 13, 1, 12, 12, 13, 1, 16, 16, 0, 16, 12, 10, 9, 1, 23, 3, 22, 0, 28, 11, 14, 23, 16, 0, 14, 6, 1, 1, 14, 4, 25, 43, 0, 29, 10, 41, 19, 47, 14, 0, 51, 10, 47, 37, 45, 46, 56, 57, 45, 10, 32, 61, 15, 9, 67, 5, 9, 22, 25, 65, 56, 24, 12, 71, 9, 57
Offset: 1

Views

Author

Benoit Cloitre, Mar 10 2002

Keywords

Examples

			log(11) = 2 + 0/2! + 2/3! + 1/4! + 2/5! + 4/6! + 3/7! + 3/8! + 1/9! + ...

Links

Crossrefs

Cf. A016634 (decimal expansion), A016739 (continued fraction).
Cf. A007514 vs. A075874 for factoradic expansion.
Cf. A067882 (log(2)), A322334 (log(3)), A322333 (log(5)), A068460 (log(7)).

Programs

  • Magma
    SetDefaultRealField(RealField(250));  [Log(11)] cat [Floor(Factorial(n)*Log(11)) - n*Floor(Factorial((n-1))*Log(11)) : n in [2..80]]; // G. C. Greubel, Dec 05 2018
  • Mathematica
    With[{b = Log[11]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 05 2018 *)
  • PARI
    vector(30, n, if(n>1, t=t%1*n, t=log(11))\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018
  • PARI
    default(realprecision, 250); b = log(11); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Dec 05 2018
  • Sage
    def a(n):
    if n==1: return floor(log(11))
    else: return expand(floor(factorial(n)*log(11)) - n*floor(factorial(n-1)*log(11)))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 05 2018

Extensions

Name edited and keyword cons,easy removed by M. F. Hasler, Nov 26 2018

A322333 Factorial expansion of log(5) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 0, 2, 3, 0, 5, 4, 4, 8, 3, 3, 2, 0, 7, 8, 0, 7, 11, 1, 18, 16, 3, 10, 16, 21, 17, 13, 20, 12, 16, 8, 27, 24, 28, 12, 9, 34, 21, 3, 9, 8, 41, 42, 35, 31, 4, 4, 37, 38, 9, 20, 10, 31, 24, 34, 44, 21, 16, 19, 24, 4, 22, 22, 47, 8, 28, 26, 32, 22, 28, 56, 44, 16, 61, 38, 3, 25, 52, 35, 73, 55, 8, 42, 25, 21, 62, 61, 7, 89, 5, 74, 89, 57, 33, 60, 13, 75, 95, 66
Offset: 1

Views

Author

G. C. Greubel, Dec 03 2018

Keywords

Examples

			log(5) = 1 + 1/2! + 0/3! + 2/4! + 3/5! + 0/6! + 5/7! + 4/8! + ...

Links

Crossrefs

Cf. A016628 (decimal expansion), A016733 (continued fraction).
Cf. A067882 (log(2)), A322334 (log(3)), A068460 (log(7)), A068461 (log(11)).

Programs

  • Magma
    SetDefaultRealField(RealField(250));  [Floor(Log(5))] cat [Floor(Factorial(n)*Log(5)) - n*Floor(Factorial((n-1))*Log(5)) : n in [2..80]];
  • Mathematica
    With[{b = Log[5]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]
  • PARI
    default(realprecision, 250); b = log(5); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))
  • Sage
    def a(n):
    if (n==1): return floor(log(5))
    else: return expand(floor(factorial(n)*log(5)) - n*floor(factorial(n-1)*log(5)))
[a(n) for n in (1..80)]
Showing 1-4 of 4 results.

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