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Numbers of the form i^j^k, for i,j,k > 1.
2

%I #13 Jan 15 2025 18:50:52

%S 16,81,256,512,625,1296,2401,4096,6561,10000,14641,19683,20736,28561,

%T 38416,50625,65536,83521,104976,130321,160000,194481,234256,262144,

%U 279841,331776,390625,456976,531441,614656,707281,810000,923521,1048576,1185921

%N Numbers of the form i^j^k, for i,j,k > 1.

%C Also numbers of the form i^j^2, since i^j^k = (i^j^(k-2))^j^2.

%H Reinhard Zumkeller, <a href="/A217261/b217261.txt">Table of n, a(n) for n = 1..10000</a>

%t t = {}; lim = 10^7; Do[n = i^j^k; If[n < lim, AppendTo[t, n]], {i, 2, lim^(1/4)}, {j, 2, Log[2, lim]}, {k, 2, Log[2, Log[2, lim]]}]; t = Union[t] (* _T. D. Noe_, Oct 01 2012 *)

%t Select[Union[Flatten[Table[i^j^2,{i,2,50},{j,2,50}]]],#<12*10^5&] (* _Harvey P. Dale_, Jan 15 2025 *)

%o (PARI) list(lim)=my(v=List());for(i=2,(.5+lim\=1)^.25, for(j=2, sqrt(log(lim+.5)/log(i)), listput(v, i^j^2))); vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Oct 01 2012

%o (Haskell)

%o import Data.Set (singleton, insert, deleteFindMin)

%o a217261 n = a217261_list !! (n-1)

%o a217261_list = f [3..] $ singleton (16, (2, 2)) where

%o f xs'@(x:xs) s

%o | m > x ^ 4 = f xs $ insert (x ^ 4, (x, 2)) s

%o | m == x ^ 4 = f xs s

%o | otherwise = m : f xs' (insert (i ^ (j + 1) ^ 2, (i, j + 1)) s')

%o where ((m, (i,j)), s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Jun 29 2013

%K nonn,nice,changed

%O 1,1

%A _Franklin T. Adams-Watters_, Sep 28 2012