A348261 Decimal expansion of the nontrivial number x for which x^Pi = Pi^x.

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

Offset

1,1

Comments

The x-th root of x equals the Pi-th root of Pi: x^(1/x) = Pi^(1/Pi) = A073238 = 1.43961949584759... .

Like Pi, is x also transcendental?

Links

Table of n, a(n) for n=1..106.

Formula

Equals -Pi*LambertW(-log(Pi)/Pi)/log(Pi). - Alois P. Heinz, Oct 13 2021

Example

2.382179087993018774555593052520878...
x^Pi = Pi^x = 15.28621734783496640312486439999472... .

Maple

evalf((t-> -LambertW(-t)/t)(log(Pi)/Pi), 120);  # Alois P. Heinz, Oct 13 2021

Mathematica

{a, b} = NSolve[x^Pi == Pi^x, x, WorkingPrecision -> 300]; a; RealDigits[N[x/.a, 300]][[1]]

Crossrefs

Cf. A000796 (Pi), A049541 (1/Pi), A073238 (Pi^(1/Pi)), A073226 (e^e, see first comment), A231737.

Sequence in context: A079585 A354854 A354861 * A252651 A058485 A204907

Adjacent sequences: A348258 A348259 A348260 * A348262 A348263 A348264

Keyword

nonn,cons

Author

Timothy L. Tiffin, Oct 08 2021

Status

approved