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A280722
Decimal expansion of the nontrivial solution to Pi^a = Pi*a.
2
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, 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, 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
OFFSET
0,1
COMMENTS
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).
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.
a satisfies (1-log(Pi))(a-1) - (a-1)^2/2 + (a-1)^3/3 - (a-1)^4/4 + (a-1)^5/5 - ... = 0
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 digits from Rok Cestnik)
Eric Weisstein's World of Mathematics, Lambert W-Function
FORMULA
Equals -LambertW(-log(Pi)/Pi)/log(Pi). - Alois P. Heinz, Mar 14 2018
EXAMPLE
0.7582711543684640692301140455152168913364...
MATHEMATICA
NSolve[Pi^a == Pi*a, a, 100]
(* Second program: *)
RealDigits[-ProductLog[-Log[Pi]/Pi]/Log[Pi], 10, 100][[1]] (* Jean-François Alcover, Oct 06 2019 *)
PROG
(PARI) solve(x=0, 0.9, Pi^x - Pi*x) \\ Michel Marcus, Jan 08 2017
CROSSREFS
Cf. A000796.
Sequence in context: A088839 A111769 A111513 * A262899 A198922 A329810
KEYWORD
nonn,cons
AUTHOR
Rok Cestnik, Jan 08 2017
STATUS
approved