A072103 Sorted perfect powers a^b for a, b > 1 with duplication.

4, 8, 9, 16, 16, 25, 27, 32, 36, 49, 64, 64, 64, 81, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 256, 256, 289, 324, 343, 361, 400, 441, 484, 512, 512, 529, 576, 625, 625, 676, 729, 729, 729, 784, 841, 900, 961, 1000, 1024, 1024, 1024, 1089

Offset

1,1

Comments

If b is the largest integer such that n=a^b for some a > 1, then n occurs d(b)-1 times in this sequence (where d = A000005 is the number of divisors function). (This includes the case where b=1 and n does not occur in the sequence.) - M. F. Hasler, Jan 25 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999, recomputed with new offset by M. F. Hasler, Jan 25 2015

Eric Weisstein's World of Mathematics, Perfect Power

Formula

Sum_{i>=2} Sum_{j>=2} 1/i^j = 1.

Example

(a,b) = (2,4) and (4,2) both yield 2^4 = 4^2 = 16, therefore 16 is listed twice.
Similarly, 64 is listed 3 times since (a,b) = (2,6), (4,3) and (8,2) all yield 64.

Maple

N:= 2000: # to get all entries <= N
sort([seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))]); # Robert Israel, Jan 25 2015

Mathematica

nn=60; Take[Sort[#[[1]]^#[[2]]&/@Tuples[Range[2, nn], 2]], nn] (* Harvey P. Dale, Oct 03 2012 *)

Prog

(Haskell)
import Data.Set (singleton, findMin, deleteMin, insert)
a072103 n = a072103_list !! (n-1)
a072103_list = f 9 3 $ Set.singleton (4, 2) where
   f zz z s
     | xx < zz   = xx : f zz z (Set.insert (x*xx, x) $ Set.deleteMin s)
     | otherwise = zz : f (zz+2*z+1) (z+1) (Set.insert (z*zz, z) s)
     where (xx, x) = Set.findMin s
-- Reinhard Zumkeller, Oct 04 2012
(PARI) is_A072103(n)=ispower(n)
for(n=1, 999, (e=ispower(n))||next; fordiv(e, d, d>1 && print1(n", "))) \\ M. F. Hasler, Jan 25 2015
(Python)
import numpy
from math import isqrt
upto = 1090
A072103 = []
for m in range(2, isqrt(upto)+1):
    k = 2
    while m**k < upto:
        A072103.append(m**k)
        k += 1
print(sorted(A072103)) # Karl-Heinz Hofmann, Sep 16 2023

Crossrefs

Cf. A001597, A000005, A253641, A253642.

Sequence in context: A329728 A141066 A018196 * A004756 A237882 A153034

Adjacent sequences: A072100 A072101 A072102 * A072104 A072105 A072106

Keyword

nonn

Author

Eric W. Weisstein, Jun 18 2002

Extensions

Offset corrected and examples added by M. F. Hasler, Jan 25 2015

Status

approved