A341328 Decimal expansion of the smaller solution (i.e., the solution other than x = 5) to 5^x = x^5.

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

Offset

1,2

Comments

Also decimal expansion of the other solution to log(x)/x = log(5)/5.

Also the limit of infinite tetration a^a^...^a, where a = 5^(1/5).

Let b be a rational number > e, then: if b is not of the form b = (1 + 1/s)^(s+1) for some positive integer s, then the other solution to b^x = x^b (or equivalently, log(x)/x = log(b)/b) is transcendental. In particular, if b is a positive integer other than 1, 2 and 4, then the other solution to b^x = x^b is transcendental (Vassilev-Missana, p. 23).

Links

Table of n, a(n) for n=1..92.

M. Vassilev-Missana, Some results on infinite power towers, Notes on Number Theory and Discrete Mathematics, Vol. 16 (2010) No. 3, 18-24.

Index entries for transcendental numbers

Formula

Equals -(5/log(5))*W(-log(5)/5), where W is the principal branch of the Lambert W function.

Example

If x = 1.7649219145257758827587235909114591014..., then log(x)/x = log(5)/5.

Mathematica

RealDigits[-5*ProductLog[-Log[5]/5]/Log[5], 10, 105]
RealDigits[x/.FindRoot[5^x==x^5, {x, 1.7}, WorkingPrecision->120], 10, 120][[1]] (* Harvey P. Dale, Jan 22 2023 *)

Prog

(PARI) default(realprecision, 92); solve(x=1, 2, 5^x-x^5)

Crossrefs

Cf. A166928.

Sequence in context: A154194 A091589 A100322 * A094961 A069814 A198816

Adjacent sequences: A341325 A341326 A341327 * A341329 A341330 A341331

Keyword

nonn,cons

Author

Jianing Song, Feb 09 2021

Status

approved