

A341328


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


0



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
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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 (VassilevMissana, p. 23).


LINKS

Table of n, a(n) for n=1..92.
M. VassilevMissana, Some results on infinite power towers, Notes on Number Theory and Discrete Mathematics, Vol. 16 (2010) No. 3, 1824.
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]


PROG

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



