A299613 -id:A299613 - cp's OEIS frontend

A299617 Decimal expansion of e^(W(1) + W(e)) = e/(W(1)*W(e)), where W is the Lambert W function (or PowerLog); see Comments.

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

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(e)) = e/(W(1)*W(e)). See A299613 for a guide to related constants.

			e^(W(1) + W(e)) = 4.7929365901428140257258473723821086015...

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

Cf. A299613, A030178.

w[x_] := ProductLog[x]; x = 1; y = E;
N[E^(w[x] + w[y]), 130]   (* A299617 *)
RealDigits[E/(LambertW[1]*LambertW[E]), 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

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

A299618 Decimal expansion of W(1) + W(1/e), where w is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Offset: 0

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 W(1) + W(1/e) = W((1/e)*(1/W(1) + 1/W(1/e))) = -1 - log(W(1)) - log(W(1/e)). See A299613 for a guide to related sequences.

			W(1) + W(1/e) = 0.845607833170857668109327401233335...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A299613, A299619.

w[x_] := ProductLog[x]; x = 1; y = 1/E; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299618 *)
RealDigits[LambertW[1] + LambertW[1/E], 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

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

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

A299624 Decimal expansion of 2*W(2), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Mar 03 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(2) = W(8/W(2)) = 2*(log(2) - log(W(2))). See A299613 for a guide to related sequences.

			2*W(2) = 1.7052110040274509826929448293906349337...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A299613, A299625.

w[x_] := ProductLog[x]; x = 2; y = 2; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299624 *)
RealDigits[2*LambertW[2],10,100][[1]] (* G. C. Greubel, Mar 03 2018 *)

2*lambertw(2) \\ G. C. Greubel, Mar 03 2018

A299632 Decimal expansion of 2*W(e/2), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Mar 13 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(e/2) = W(e^2/(2*W(e/2))) = 2 - log(4) - 2*log(W(e/2)). See A299613 for a guide to related sequences.

			2*W(e/2) = 1.3701538843091878920564989610752603...

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A299613, A299632.

w[x_] := ProductLog[x]; x = E/2; y = E/2; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299632 *)

2*lambertw(exp(1)/2) \\ Altug Alkan, Mar 13 2018

A299614 Decimal expansion of e^(2A030178) = e^(2W(1)) = (1/W(1))^2, where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

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^(2*W(1)) = (W(1))^(-2). See A299613 for a guide to related constants.

			e^(2*W(1)) = 3.1089547635799361854809454054...

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

Cf. A299613.

w[x_] := ProductLog[x]; x = 1; y = 1;
N[E^(w[x] + w[y]), 130]   (* A299614 *)
RealDigits[(1/LambertW[1])^2, 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

(1/lambertw(1))^2 \\ G. C. Greubel, Mar 03 2018

A299615 Decimal expansion of W(1) + W(2), where w is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

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 W(1) + W(2) = W(2/W(1) + 2/W(2)) = log(2) - log(W(1)) - log(W(2)). See A299613 for a guide to related sequences.

			W(1) + W(2) = 1.41974879242350936434644...

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

Cf. A299613, A299616.

w[x_] := ProductLog[x]; x = 1; y = 2; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299615 *)
RealDigits[LambertW[1] + LambertW[2], 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

lambertw(1) + lambertw(2) \\ G. C. Greubel, Mar 03 2018

A299616 Decimal expansion of e^(W(1) + W(2)) = 2/(W(1)*W(2)), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

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(2)) = 2/(W(1)*W(2)). See A299613 for a guide to related constants.

			e^(W(1) + W(2)) = 4.13608129477801994342...

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

Cf. A299613, A299615.

w[x_] := ProductLog[x]; x = 1; y = 2; N[E^(w[x] + w[y]), 130]   (* A299616 *)
RealDigits[2/(LambertW[1]*LambertW[2]), 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

2/(lambertw(1)*lambertw(2)) \\ G. C. Greubel, Mar 03 2018

A299620 Decimal expansion of W(1) + W(3), where w is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

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 W(1) + W(3) = W(3/W(1) + 3/W(3)) = log(3) - log(W(1)) - log(W(3)). See A299613 for a guide to related sequences.

			W(1) + W(3) = 1.6170521853738238329886657327632...

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

Cf. A299613, A299621.

w[x_] := ProductLog[x]; x = 1; y = 3; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299620 *)
RealDigits[LambertW[1] + LambertW[3], 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

lambertw(1) + lambertw(3) \\ G. C. Greubel, Mar 03 2018

A299621 Decimal expansion of e^(W(1) + W(3)) = 3/(W(1)*W(3)), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

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(3)) = 3/(W(1)*W(3)). See A299613 for a guide to related constants.

			e^(W(1) + W(3)) = 5.03821667615918674918541702644888...

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

Cf. A299613, A299620.

w[x_] := ProductLog[x]; x = 1; y = 3;
N[E^(w[x] + w[y]), 130]   (* A299621 *)
RealDigits[3/(LambertW[1]*LambertW[3]), 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

3/(lambertw(1)*lambertw(3)) \\ G. C. Greubel, Mar 03 2018

cp's OEIS Frontend

A299617 Decimal expansion of e^(W(1) + W(e)) = e/(W(1)*W(e)), where W is the Lambert W function (or PowerLog); see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A299618 Decimal expansion of 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

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

A299624 Decimal expansion of 2*W(2), where W is the Lambert W function (or PowerLog); see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A299632 Decimal expansion of 2*W(e/2), where W is the Lambert W function (or PowerLog); see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A299614 Decimal expansion of e^(2*A030178) = e^(2*W(1)) = (1/W(1))^2, where W is the Lambert W function (or PowerLog); see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A299615 Decimal expansion of W(1) + W(2), where w is the Lambert W function (or PowerLog); see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A299616 Decimal expansion of e^(W(1) + W(2)) = 2/(W(1)*W(2)), where W is the Lambert W function (or PowerLog); see Comments.

Views

A299614 Decimal expansion of e^(2A030178) = e^(2W(1)) = (1/W(1))^2, where W is the Lambert W function (or PowerLog); see Comments.