%I #49 Oct 06 2019 07:47:00
%S 7,5,8,2,7,1,1,5,4,3,6,8,4,6,4,0,6,9,2,3,0,1,1,4,0,4,5,5,1,5,2,1,6,8,
%T 9,1,3,3,6,4,2,1,1,0,8,7,6,3,4,1,4,6,2,2,1,4,9,9,7,2,1,0,0,1,4,6,9,6,
%U 0,8,1,6,7,0,2,6,7,8,0,4,0,8,3,8,5,1,1,0,4,4,3,9,6,4,6,3,3,5,9,0
%N Decimal expansion of the nontrivial solution to Pi^a = Pi*a.
%C Expressions A^a = A*a for A > 1 have two real solutions, one of which is trivial a = 1 (the exception being A = e = 2.718... where both solutions coincide).
%C Infinite expressions for a can arise from Pi^a = Pi*a, such as, a = Pi^(Pi^(Pi^(...)-1)-1) and a = log_Pi(Pi*log_Pi(Pi*log_Pi(...))), however, they are by themselves not well defined, because the values of their partial expressions do not converge or converge to different values depending on how the expression is truncated.
%C a satisfies (1-log(Pi))(a-1) - (a-1)^2/2 + (a-1)^3/3 - (a-1)^4/4 + (a-1)^5/5 - ... = 0
%H Alois P. Heinz, <a href="/A280722/b280722.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 digits from Rok Cestnik)
%H Rok Cestnik, <a href="/A280722/a280722.pdf">Plot of the difference Pi^a-Pi*a</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F Equals -LambertW(-log(Pi)/Pi)/log(Pi). - _Alois P. Heinz_, Mar 14 2018
%e 0.7582711543684640692301140455152168913364...
%t NSolve[Pi^a == Pi*a, a, 100]
%t (* Second program: *)
%t RealDigits[-ProductLog[-Log[Pi]/Pi]/Log[Pi], 10, 100][[1]] (* _Jean-François Alcover_, Oct 06 2019 *)
%o (PARI) solve(x=0, 0.9, Pi^x - Pi*x) \\ _Michel Marcus_, Jan 08 2017
%Y Cf. A000796.
%K nonn,cons
%O 0,1
%A _Rok Cestnik_, Jan 08 2017