A217261 Numbers of the form i^j^k, for i,j,k > 1.

16, 81, 256, 512, 625, 1296, 2401, 4096, 6561, 10000, 14641, 19683, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 262144, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1048576, 1185921, 1336336, 1500625, 1679616, 1874161, 1953125

Offset: 1

Author

Franklin T. Adams-Watters, Sep 28 2012

Keywords

• nonn
• nice

Comments

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Programs

Haskell

import Data.Set (singleton, insert, deleteFindMin)
a217261 n = a217261_list !! (n-1)
a217261_list = f [3..] $ singleton (16, (2, 2)) where
   f xs'@(x:xs) s
     | m > x ^ 4  = f xs $ insert (x ^ 4, (x, 2)) s
     | m == x ^ 4 = f xs s
     | otherwise  = m : f xs' (insert (i ^ (j + 1) ^ 2, (i, j + 1)) s')
     where ((m, (i,j)), s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 29 2013

Mathematica

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

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

Python

def generate(limit: int) -> list:
    numbers = set()
    # For very large range this list must be expanded:
    for n in [2, 3, 5, 7]:
        m = 2
        j = n * n
        while m**j <= limit:
            numbers.add(m**j)
            m += 1
    return sorted(numbers)
A217261List = generate(2000000)
print(A217261List)  # Peter Luschny, May 07 2025