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.

A299613 Decimal expansion of 2*W(1), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 01 2018

Keywords

Comments

The Lambert W function satisfies the functional equations
W(x) + W(y) = W(x*y*(1/W(x) + 1/W(y))) = log(x*y)/(W(x)*W(y)) for x and y greater than -1/e, so that 2*W(1) = W(2/W(1)) = -2*log(W(1)).
Guide to related constants:
--------------------------------------------
x y W(x) + W(y) e^(W(x) + W(y))
--------------------------------------------
1 1 A299613 A299614
1 2 A299615 A299616
1 e A030178 A299617
e e 2 exactly e^2 exactly
1 1/e A299618 A299619
1 3 A299620 A299621
1 1/2 A299622 A299623
1/2 1/2 A126583 A099954
2 2 A299624 A299625
3 3 A299626 A299627
1/3 1/3 A299628 A299629
3/2 3/2 A299630 A299631
e/2 e/2 A299632 A299633

Examples

			2*W(1) = 1.13428658081956774599993...

Links

Crossrefs

Cf. A299614-A299633.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 1; y = 1; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299613 *)
RealDigits[2 ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v, Mar 02 2018 *)
  • PARI
    2*lambertw(1) \\ G. C. Greubel, Mar 07 2018

Formula

Equals 2*A030178.

A201936 Decimal expansion of the least number x satisfying 2*x^2=e^(-x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 13 2011

Keywords

Comments

For some choices of a, b, c, there is a unique value of x satisfying a*x^2+bx+c=e^x; for other choices, there are two solutions; and for others, three. Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 0.... 0.... A126583
2.... 0.... 0.... A201936, A201937, A201938
1.... 0... -1.... A201940
1.... 1.... 0.... A201941
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A201936, take f(x,u,v)=u*x^2+v-e^(-x) and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

Examples

			least x:  -2.617866613066812769178978059143202...
greatest negative x:  -1.487962065498177156254...
greatest x:  0.5398352769028200492118039083633...

Links

Crossrefs

Cf. A201741 [a*x^2+b*x+c=e^x].

Programs

  • Mathematica
    a = 2; b = 0; c = 0;
f[x_] := a*x^2 + b*x + c; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -3, -2}, WorkingPrecision -> 110]
RealDigits[r]  (* A201936 *)
r = x /. FindRoot[f[x] == g[x], {x, -2, -1}, WorkingPrecision -> 110]
RealDigits[r]   (* A201937 *)
r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]   (* A201938 *)
(* Program 2: implicit surface of u*x^2+v=e^(-x) *)
f[{x_, u_, v_}] := u*x^2 + v - E^-x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, -4, 0}, {u, 1,10}];
ListPlot3D[Flatten[t, 1]]  (* for A201936 *)

A387101 Decimal expansion of the smallest real solution to e^x = x^3.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, Aug 16 2025

Keywords

Comments

Equivalently, the smallest real solution to log(x) = x/3.

Examples

			1.85718386020783533645698098206276669990441533...

Links

Crossrefs

Cf. A030178, A126583, A126584, A387102 (largest).

Programs

  • Mathematica
    RealDigits[-3*ProductLog[-1/3],10,100][[1]]
RealDigits[x/.FindRoot[E^x==x^3,{x,1},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Sep 02 2025 *)
  • PARI
    -3*lambertw(-1/3) \\ Michel Marcus, Aug 18 2025

Formula

Equals -3*LambertW(-1/3).

A387102 Decimal expansion of the largest real solution to e^x = x^3.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, Aug 16 2025

Keywords

Comments

Equivalently, the largest real solution to log(x) = x/3.

Examples

			4.5364036549735274216902190342161161138109511554...

Links

Crossrefs

Cf. A030178, A126583, A126584, A366565, A387101 (smallest).

Programs

  • Mathematica
    RealDigits[-3*ProductLog[-1,-1/3],10,100][[1]]

Formula

Equals -3*LambertW(-1, -1/3).
Showing 1-4 of 4 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.