This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348359 #18 Dec 09 2024 16:13:42
%S A348359 6,0,5,5,7,2,2,0,9,1,0,2,4,7,4,1,0,0,2,1,2,6,6,3,9,1,1,7,5,8,3,1,4,9,
%T A348359 7,3,1,6,8,3,8,2,8,7,5,3,7,8,3,6,7,7,7,4,3,9,4,9,9,6,7,7,3,5,2,8,1,8,
%U A348359 7,9,7,4,4,8,5,2,3,5,8,1,4,7,9,3,8,9,4,6,6,6,0,7,4,2,8,1,7,8,9,4,7,8,9,4,5,7
%N A348359 Decimal expansion of the nontrivial number x for which x^phi = phi^x, where phi is the golden ratio (1+sqrt(5))/2.
%C A348359 The x-th root of x equals the phi-th root of phi: x^(1/x) = phi^(1/phi) = A185261 = 1.3463608200348694434247534661858... .
%C A348359 Not surprisingly, x appears to be irrational. If x is also algebraic, then x^phi would be transcendental by the Gelfond-Schneider theorem.
%H A348359 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem">Gelfond-Schneider theorem</a>
%e A348359 6.055722091024741002126639117583149731683828...
%e A348359 x^phi = phi^x = 18.431940924839652158136364051482054378959672... .
%t A348359 RealDigits[x/.FindRoot[x^GoldenRatio==GoldenRatio^x,{x,6},WorkingPrecision->120],10,120][[1]] (* _Harvey P. Dale_, Dec 09 2024 *)
%Y A348359 Cf. A001622 (phi), A094214 (1/phi), A185261 (phi^(1/phi)), A073226 (e^e, see first comment).
%K A348359 nonn,cons
%O A348359 1,1
%A A348359 _Timothy L. Tiffin_, Oct 14 2021
%E A348359 Prior Mathematica program replaced by _Harvey P. Dale_, Dec 09 2024