A299618 -id:A299618 - cp's OEIS frontend

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

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

Clark Kimberling, Mar 01 2018

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

			2*W(1) = 1.13428658081956774599993...

Clark Kimberling, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lambert W-Function.
Index entries for sequences related to LambertW function.
Index entries for transcendental numbers.

Cf. A299614-A299633.

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 *)

2*lambertw(1) \\ G. C. Greubel, Mar 07 2018

Equals 2*A030178.

A299619 Decimal expansion of e^(W(1) + W(1/e)) = (1/e)/(W(1)*W(1/e)), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 01 2018

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(W(1) + W(1/e)) = (1/e)/(W(1)*W(1/e)). See A299613 for a guide to related constants.

			e^(W(1) + W(1/e)) = 2.3293932668427932248576309...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A299613, A299618.

w[x_] := ProductLog[x]; x = 1; y = 1/E;
N[E^(w[x] + w[y]), 130]   (* A299619 *)
RealDigits[1/(E*LambertW[1]*LambertW[1/E]), 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

exp(-1)/(lambertw(1)*lambertw(exp(-1))) \\ G. C. Greubel, Mar 03 2018

A375596 Decimal expansion of 1/(1 - W(1/e)), where W is the Lambert W function.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Aug 20 2024

			1.38593327599819425386062181488251586064513...

Edgar Assing, On gaps between primes. See p. 45.

Cf. A202357.

Cf. A141606, A299617, A299618, A299619, A299632, A299633, A307951, A322166.

RealDigits[1/(1-ProductLog[1/E]),10,100][[1]]

1/(1-lambertw(1/exp(1))) \\ Michel Marcus, Aug 20 2024

Equals 1/(1 - A202357).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A299619 Decimal expansion of e^(W(1) + W(1/e)) = (1/e)/(W(1)*W(1/e)), where W is the Lambert W function (or PowerLog); see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A375596 Decimal expansion of 1/(1 - W(1/e)), where W is the Lambert W function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula