I saw a picture pointing out that sqrt(2^6^2^1^4^4) = 262144, so I wanted to write a Guile program to find other such numbers:
(use-modules ((rnrs) :version (6) :select (div-and-mod))
(srfi srfi-11)
(srfi srfi-41))
;; Go through the digits of a positive integer, right-to-left.
(define (fold-digits f n a)
(cond
((> 10 n) (f a n))
(else
(let-values (((rest digit) (div-and-mod n 10)))
(fold-digits f rest (f a digit))))))
(define (special? n)
(= (sqrt (fold-digits (lambda (x y) (expt y x)) n 1)) n))
(display
(stream->list 3 (stream-filter special? (stream-from 0))))
(newline)But it quickly gets stuck at 268 as calculating 2^(6^8)=2^1679616 is taking a bit too long... Now I am thinking about how to avoid this and other impossible calculations.