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 1-5 of 5 results.

A119824 Decimal expansion of area bounded by x->Exp[x] and x->Gamma[x+1] on 0 <= x <= c, where c is the value given by A078335.

Original entry on oeis.org

7, 4, 9, 0, 3, 1, 4, 2, 2, 5, 8, 2, 7, 6, 0, 5, 8, 3, 0, 2, 6, 3, 3, 0, 4, 7, 5, 9, 1, 3, 6, 8, 9, 8, 4, 6, 5, 3, 7, 1, 1, 3, 1, 0, 2, 1, 3, 1, 1, 6, 0, 1, 0, 6, 1, 0, 9, 1, 6, 4, 9, 3, 4, 3, 6, 1, 3, 3, 0, 2, 6, 8, 4, 1, 6, 2, 3, 8, 1, 2, 3, 2, 7, 8, 0, 3, 1, 6, 6, 0, 4, 4, 6, 9, 8, 6, 5, 9, 2, 5, 0, 8, 4, 7, 3, 4
Offset: 2

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 30 2006

Keywords

Examples

			74.9031422582760583026330475914...

Crossrefs

Cf. Upper bound of integral is given by A078335.

Programs

  • Mathematica
    RealDigits[NIntegrate[Exp[x] - Gamma[x + 1], {x, 0, FindRoot[r - Log[r! ], {r, 5.29}, WorkingPrecision -> 200][[1]][[2]]}, WorkingPrecision -> 40]]

Extensions

More terms from Robert G. Wilson v, Nov 08 2013

A119858 Decimal expansion of area bounded by x->x and x->Log[x! ] on 0 <= x <= c, where c is the value given by A078335.

Original entry on oeis.org

4, 6, 6, 2, 0, 1, 6, 7, 0, 3, 1, 8, 2, 1, 1, 1, 3, 4, 0, 9, 1, 1, 0, 7, 0, 3, 6, 4, 1, 2, 4, 0, 8, 6, 8, 5, 1, 3, 5, 5, 9, 8, 1, 8, 6, 2, 0, 1, 7, 7, 8, 3, 7, 0, 2, 9, 1, 1, 4, 2, 1, 3, 0, 6, 2, 8, 9, 4, 6, 2, 2, 5, 7, 1, 1, 2, 1, 4, 7, 1, 2, 0, 6, 0, 4, 1, 2, 0, 7, 5, 5, 0, 5, 5, 7, 1, 9, 4, 7, 2, 9, 9, 0, 8, 2
Offset: 1

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 30 2006

Keywords

Examples

			4.66201670318211134091107036412...

Crossrefs

Cf. Upper bound of integral is given by A078335.

Programs

  • Mathematica
    RealDigits[NIntegrate[x - Log[x! ], {x, 0, FindRoot[r - Log[r! ], {r,5.29}, WorkingPrecision -> 200][[1]][[2]]}, WorkingPrecision -> 40]]
  • PARI
    intnum(x=0, solve(x=5.2, 5.3, exp(x)-gamma(1+x)) , x - log(gamma(x+1))) \\ Michel Marcus, Apr 05 2015

Extensions

More terms from Robert G. Wilson v, Jan 07 2016

A330380 Decimal expansion of the y-coordinate for the largest solution to e^x = Gamma(x+1).

Original entry on oeis.org

1, 9, 8, 4, 0, 6, 1, 3, 0, 3, 1, 1, 2, 7, 6, 7, 7, 6, 9, 1, 1, 5, 0, 4, 2, 7, 2, 6, 0, 1, 9, 4, 7, 4, 8, 1, 5, 6, 2, 4, 2, 3, 0, 3, 8, 2, 9, 8, 2, 5, 2, 8, 7, 6, 3, 0, 6, 6, 9, 6, 4, 9, 9, 2, 2, 5, 5, 8, 3, 4, 1, 7, 4, 3, 3, 8, 8, 7, 6, 1, 2, 7, 3, 4, 9, 9, 1, 3, 6, 1, 2, 4, 8, 3, 3, 9, 3, 1, 1, 2, 8, 5, 1, 0, 4, 0, 4, 4, 9, 2, 4, 4, 7, 2, 3
Offset: 3

Views

Author

Eliora Ben-Gurion, Dec 12 2019

Keywords

Comments

This number is the y-coordinate of the point at which the factorial function, Gamma(x+1), begins to exceed the exponential function.

Examples

			x = 5.29031609311977071072...
y = 198.40406130311276776911...

Links

Crossrefs

Cf. A078335.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[Gamma[Log[x] + 1] == x, {x, 200}, WorkingPrecision -> 120], 10, 115][[1]] (* Amiram Eldar, May 31 2021 *)
  • PARI
    \p200
exp(solve (x=5,6,exp(x)-gamma(x+1))) \\ Hugo Pfoertner, Dec 12 2019

Formula

Equals exp(A078335).

A336763 Decimal expansion of largest real root of e^x = Gamma(x).

Original entry on oeis.org

7, 4, 6, 3, 6, 0, 3, 2, 8, 3, 7, 7, 6, 4, 4, 6, 1, 3, 7, 8, 1, 0, 5, 0, 1, 0, 9, 5, 8, 4, 5, 3, 1, 8, 5, 8, 9, 6, 1, 5, 4, 8, 8, 6, 3, 6, 5, 1, 3, 8, 4, 2, 5, 4, 4, 5, 6, 2, 5, 4, 3, 7, 4, 2, 8, 5, 4, 3, 2, 0, 2, 5, 1, 8, 8, 3, 3, 1, 7, 3, 9, 4, 1, 9, 1, 5, 2, 2, 6, 8, 6, 5, 0, 1, 6, 3, 9, 6, 1, 4, 8, 6, 3, 7
Offset: 1

Views

Author

Daniel Hoyt, Aug 03 2020

Keywords

Comments

This number corresponds to the point where the Gamma function starts to exceed the exponential function.

Examples

			x = 7.46360328377644613...

Crossrefs

Cf. A078335.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[Exp[x] == Gamma[x], {x, 7}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 11 2023 *)
  • PARI
    solve(x=7.4, 7.5, exp(x)-gamma(x))

A338460 Decimal expansion of the largest real root of e^(x-1) = Gamma(x+1).

Original entry on oeis.org

3, 6, 1, 4, 7, 9, 3, 7, 0, 3, 1, 9, 2, 5, 2, 5, 4, 4, 7, 3, 8, 6, 5, 3, 6, 6, 2, 5, 6, 0, 3, 4, 5, 4, 6, 3, 3, 5, 3, 1, 5, 1, 6, 5, 9, 6, 9, 4, 7, 5, 0, 2, 2, 6, 6, 1, 1, 1, 5, 9, 9, 9, 7, 7, 4, 6, 2, 5, 1, 8, 2, 9, 8, 6, 1, 3, 6, 1, 8, 5, 7, 5, 4, 3, 2, 8, 1, 8, 6, 2, 8, 2, 1, 5, 7, 1, 1, 5, 9, 6, 3, 3, 0, 8, 1
Offset: 1

Views

Author

Michael P. May, Jan 31 2021

Keywords

Comments

Decimal expansion of the constant value for which the average and the minimum prime gaps are equal for the prime number sequences of higher order P', P'', P''', and P'''' as represented by A333242, A262275, A333243 and A333244.
x-1 is the smallest average gap size for the set of all prime numbers P. - Michael P. May, Jan 26 2025

Examples

			3.61479370319252544738653662560345463353151659694750...

Crossrefs

Cf. A333242, A262275, A333243, A333244.
Cf. A001113, A078335, A336763.

Programs

  • Maple
    Digits:= 155:
fsolve(exp(x-1)=GAMMA(x+1), x=3..4);  # Alois P. Heinz, Feb 01 2021
  • Mathematica
    RealDigits[x /. FindRoot[LogGamma[x + 1] - x + 1, {x, 3}, WorkingPrecision -> 110], 10, 105][[1]] (* Amiram Eldar, Feb 01 2021 *)
  • PARI
    solve(x=3,4,lngamma(x+1)-x+1) \\ Hugo Pfoertner, Feb 01 2021

Formula

x| (log(x!))^n * (log(x!) + 1) = x * (x-1)^n, for n >= 0
Showing 1-5 of 5 results.

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