A300916 -id:A300916 - cp's OEIS frontend

A277522 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/2.

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

Offset: 0

Alois P. Heinz, Oct 19 2016

			0.5692452044263480610653304778419669052638659731463...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277523, A277524, A277525, A277526, A277527, A277528, A277529, A277530, A277531, A277559, A277651, A293009, A300916.

RealDigits[2 Exp[-2 ProductLog[Log[2]]]/(1 + ProductLog[Log[2]]), 10, 105][[1]] (* Vladimir Reshetnikov, Oct 20 2016 *)
f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f'[1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

2*exp(-2*lambertw(log(2)))/(1+lambertw(log(2))) \\ G. C. Greubel, Nov 10 2017

Equals 2*exp(-2*LambertW(log(2)))/(1+LambertW(log(2))). - Vladimir Reshetnikov, Oct 20 2016

A277651 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/4.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Oct 25 2016

It is known that (1/4)^(1/4)^(1/4)^... = 1/2.

			0.59061610914964124974380690932325155711665304887388...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A002162, A277522, A277559, A293009, A300916.

RealDigits[1/(1 + Log[2]), 10, 120][[1]]
(* or *)
f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f'[1/4], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

Equals 1/(1 + log(2)) = 1/(1 + A002162).

Equals Integral_{x >= 0} exp(-x)/2^x dx. - Peter Bala, Feb 05 2024

A293009 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/Pi.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Mar 16 2018

			0.56501844596024150528994096062245192028392680078511838285519...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration

Cf. A000796, A049541, A073243, A277522, A277651, A300916.

RealDigits[Pi*Exp[-2*LambertW[Log[Pi]]]/(1+LambertW[Log[Pi]]), 10, 100][[1]] (* G. C. Greubel, Sep 09 2018 *)

Pi*exp(-2*lambertw(log(Pi)))/(1+lambertw(log(Pi))) \\ Michel Marcus, Mar 16 2018

Equals Pi*exp(-2*LambertW(log(Pi)))/(1+LambertW(log(Pi))).

cp's OEIS Frontend

A277522 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/2.

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A277651 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/4.

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A293009 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/Pi.

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