

A072103


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


13



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
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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



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 !! (n1)
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
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
for m in range(2, isqrt(upto)+1):
k = 2
while m**k < upto:
k += 1


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



