A077590 -id:A077590 - cp's OEIS frontend

A049006 Decimal expansion of i^i = exp(-Pi/2).

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

Offset: 0

Deepak R. N (deepak_rama(AT)bigfoot.com)

Equals 1/A042972. - Lekraj Beedassy, Sep 02 2005
Euler knew this number to be purely real, and called the fact "remarkable" in a letter to Goldbach dated June 14, 1746. - Alonso del Arte, Nov 30 2012
The value follows immediately from Euler's formula i = exp(i Pi/2) and the rule (a^b)^c = a^(b*c). - The value given by Uhler has the final digits ...14 instead ...08, which is compatible with the claimed accuracy of 52 digits. - M. F. Hasler, May 17 2018

			0.20787957635076190854695561983497877003387...

Florian Cajori, History of Mathematics. New York: Chelsea Publishing Company for the American Mathematical Society (1991): 236.
Ian Connell, Modern Algebra: A Constructive Introduction. New York: Elsevier (1981) p. 363.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, page 97.
Reinhold Remmert, Theory of Complex Functions: Readings in Mathematics. New York: Springer-Verlag (1991): 162.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Leonhard Euler, Letter to Christian Goldbach, Berlin, June 14 1746.
Simon Plouffe, exp(-Pi/2) also i**i to 10000 digits.
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Eric Weisstein's World of Mathematics, i.
Index entries for transcendental numbers.

Cf. A042972, A049007, A097665, A202501 (tetration).

Cf. A077589 and A077590 for i^i^i^...

Mathematica
```
RealDigits[Re[N[I^I, 100]]][[1]]
```

{ default(realprecision, 20080); x=10*exp(-Pi/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b049006.txt", n, " ", d)); } \\ Harry J. Smith, Apr 28 2009, corrected May 19 2009

digits(exp(-Pi/2)\.1^default(realprecision))[^-1] \\ M. F. Hasler, May 17 2018

Equals 1/A042972 = 2*A097665. - Hugo Pfoertner, Aug 21 2024

A198683 Number of distinct values taken by i^i^...^i (with n i's and parentheses inserted in all possible ways) where i = sqrt(-1) and ^ denotes the principal value of the exponential function.

Original entry on oeis.org

1, 1, 2, 3, 7, 15, 34, 77, 187, 462, 1152

Offset: 1

Vladimir Reshetnikov, Oct 28 2011

There are C(n-1) ways of inserting the parentheses (where C is a Catalan number, A000108), but not all arrangements produce different values.
At n=10, the expression i^(i^(((i^i)^i)^(i^((i^i)^(i^i))))) evaluates to a large complex number, C = -6.795047376...*10^34 - i*6.044219499...*10^34; as a result, i^C, which arises at n=11, is very large, having a magnitude of e^((-Pi/2)*(-6.044219499...*10^34)) = 4.1007...*10^41232950809707420597749203381002924. - Jon E. Schoenfield, Nov 21 2015
Note that if a is a REAL positive number, the number of different values of a^a^...^a with n a's is at most A000081(n). But this relies on the identity (x^y)^z = (x^z)^y = x^(yz), which is not always true for complex numbers with the principal value of the power function. Thus if Y = ((i^i)^i)^i, we have (i^i)^Y / (i^Y)^i = exp(-2 Pi). - Robert Israel, Nov 27 2015 [So for the present sequence, we know a(n) <= A000108(n-1), but we do not know that a(n) <= A000081(n). - N. J. A. Sloane, Nov 28 2015]

			a(1) = 1: there is one value, i.
a(2) = 1: there is one value, i^i = exp(i Pi / 2)^i = exp(-Pi/2) = 0.2079...
a(3) = 2: there are two values: (i^i)^i = i^(-1) = 1/i = -i and i^(i^i) = i^0.2079... = exp(0.2079... i Pi / 2) = 0.9472... + 0.3208... i.
a(4) = 3: there are 5 possible parenthesizations but they give only 3 distinct values: i^(i^(i^i)), i^((i^i)^i) = ((i^i)^i)^i, (i^i)^(i^i) = (i^(i^i))^i.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
MathOverflow, Discussion of related questions
Jon E. Schoenfield, Tables for n = 1..11 listing all A000108(n-1) ways of inserting the parentheses and identifying the ways that do not yield duplicated values

Cf. A000081, A000108, A002845, A049006, A077589, A077590.

iParens[1] = {I}; iParens[n_] := iParens[n] = Union[Flatten[Table[Outer[Power, iParens[k], iParens[n - k]], {k, n - 1}]], SameTest -> Equal]; Table[Length[iParens[n]], {n, 10}]

a(11) and a(12) (unconfirmed) from Alonso del Arte, Nov 17 2011

a(12) is said to be either 2919 or 2926. The value will not be included in the data section until it has been confirmed. - N. J. A. Sloane, Nov 26 2015

A077589 Decimal expansion of real part of the infinite power tower of i.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 07 2002

This is the real part of i^i^i^i^i^i...

			0.43828293672703211162697516355126482426789735164639460360922124049579153222269568...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

Eric Weisstein's World of Mathematics, i.
Eric Weisstein's World of Mathematics, Power Tower.

Cf. A049006, A077590 (imaginary part).

evalf(Re(2*I*LambertW(-I*Pi/2)/Pi), 137);  # Alois P. Heinz, Dec 12 2023

Prepend@@RealDigits[Re[ -ProductLog[ -Log[I]]/Log[I]], 10, 150]

z=(1+I)/2;e=.1^default(realprecision);until(e>abs(z-z-=(z-I^z)/(1-I^(z+1)*Pi/2)),);digits(real(z)\e) \\ M. F. Hasler, May 17 2018

The value is 2 (i/Pi) W(-i Pi/2) = 0.4382829... + i 0.360592..., where W denotes the principal branch of the Lambert W function. - David W. Cantrell, Nov 23 2007

A305200 Decimal expansion of the real part of continued exponential of i.

Original entry on oeis.org

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

Offset: 0

Keerthi Vasan Gopala, May 27 2018

			0.576412723031435283148289239887068476278...

This is the real part of e^(i*e^(i*e^(i...))).

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

Cf. A049006, A077589, A077590, A191719, A305202.

RealDigits[Re[I*LambertW[-I]],10,120][[1]] (* Harvey P. Dale, Dec 01 2018 *)
RealDigits[x /. FindRoot[E^(x*Tan[x]) == Cos[x]/x, {x, 1/2}, WorkingPrecision -> 120]][[1]] (* Vaclav Kotesovec, Oct 02 2021 *)

Equals Re(i*LambertW(-i)). - Alois P. Heinz, May 27 2018
From Vaclav Kotesovec, Oct 02 2021: (Start)
Root of the equation exp(x*tan(x)) = cos(x)/x.
Equals Im(LambertW(i)). (End)

More digits from Alois P. Heinz, May 27 2018

A100120 Limit of the power tower t(2)^(t(3)^(t(5)^(t(7)^(...)))) with t(n)=n!^(1/n!) and n taking prime values.

Original entry on oeis.org

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 09 2004

Let f(n)=n!^(1/n!). Then this number is f(2)^(f(3)^(f(5)^(...)))

			1.604651218410055827983866511015173398808649754699580...

Cf. A077589, A077590, A073243, A098454, A098584.

default(realprecision,100):t=1:forstep(n=100,1,-1,t=(prime(n)!^(1/prime(n)!))^t):return(t)

Offset corrected by R. J. Mathar, Feb 05 2009

A212479 Decimal expansion of the absolute value of infinite power tower of i.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 17 2012

This c = |z|, where z is the complex solution of z = i^z or, equivalently, z = i^i^i^...

			0.5675551633069578253846131441924533439 ...

Eric Weisstein's World of Mathematics, Power Tower

Cf. A077589 (real part of z), A077590 (imaginary part of z), A212480 (argument of z).

2*I*ProductLog[-I*Pi/2]/Pi // Abs // N[#, 108]& // RealDigits[#][[1]]& (* Jean-François Alcover, Feb 05 2013 *)

my(z="I"); for (i=1, 1000, z = concat(z, "^I")); z = eval(z); sqrt(norml2([real(z), imag(z)])) \\ Michel Marcus, May 12 2023

c = |i^i^i^...|.

A212480 Decimal expansion of the argument of infinite power tower of i.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 17 2012

This c, expressed in radians, equals arg(z), where z is the complex solution of z = i^z or, equivalently, z = i^i^i^... Also, c = atan(A077590/A077589).

			0.6884532271077021304987675711768242596 ...

Eric Weisstein's World of Mathematics, Power Tower

Cf. A077589 (real part of z), A077590 (imaginary part of z), A212479 (absolute value of z).

2*I*ProductLog[-I*Pi/2]/Pi // Arg // N[#, 108]& // RealDigits[#][[1]]& (* Jean-François Alcover, Feb 05 2013 *)

\\ start with I^(0.4+0.4*I) and iterate (%+I^%)/2. It converges pretty rapidly to z.

c = arg(i^i^i^...).

A305208 Decimal expansion of the real part of the continued exponential i/Pi.

Original entry on oeis.org

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

Offset: 0

Keerthi Vasan Gopala, May 27 2018

This is the real part of e^((i/Pi)*e^((i/Pi)*e^((i/Pi)...))).

			0.88530292263172060173561116234106499518957753397967...

Cf. A049006, A077589, A077590, A305200, A305202, A305210.

Mathematica
```
Re[Pi*I*N[ProductLog[-I/Pi], 100]]
```

Equals Re(Pi*i*LambertW(-i/Pi)).

A305210 Decimal expansion of the imaginary part of continued exponential (i/Pi).

Original entry on oeis.org

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

Offset: 0

Keerthi Vasan Gopala, May 27 2018

This is the imaginary part of e^((i/Pi)*e^((i/Pi)*e^((i/Pi)...))).

			0.256299537163861312529996729880982538078341463884...

Cf. A049006, A077589, A077590, A305200, A305202, A305208.

Mathematica
```
Im[Pi*I*N[ProductLog[-I/Pi], 100]]
```

Equals Im(Pi*i*LambertW(-i/Pi)).

cp's OEIS Frontend

A049006 Decimal expansion of i^i = exp(-Pi/2).

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A198683 Number of distinct values taken by i^i^...^i (with n i's and parentheses inserted in all possible ways) where i = sqrt(-1) and ^ denotes the principal value of the exponential function.

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A077589 Decimal expansion of real part of the infinite power tower of i.

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Maple

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A305200 Decimal expansion of the real part of continued exponential of i.

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Extensions

A100120 Limit of the power tower t(2)^(t(3)^(t(5)^(t(7)^(...)))) with t(n)=n!^(1/n!) and n taking prime values.

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PARI

Extensions

A212479 Decimal expansion of the absolute value of infinite power tower of i.

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Mathematica

PARI

Formula

A212480 Decimal expansion of the argument of infinite power tower of i.

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