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A072103 Sorted perfect powers a^b for a, b > 1 with duplication. 5
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 (list; graph; refs; listen; history; text; internal format)
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}^{infty} sum_{j=2}^{infty} 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) \\ M. F. Hasler, Jan 25 2015

for(n=1, 999, (e=ispower(n))||next; fordiv(e, d, d>1 && print1(n", ")) \\ M. F. Hasler, Jan 25 2015

CROSSREFS

Cf. A001597, A000005, A253641, A253642.

Sequence in context: A020217 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

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Last modified July 8 00:21 EDT 2015. Contains 259363 sequences.