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A097054
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Nonsquare perfect powers.
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10
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8, 27, 32, 125, 128, 216, 243, 343, 512, 1000, 1331, 1728, 2048, 2187, 2197, 2744, 3125, 3375, 4913, 5832, 6859, 7776, 8000, 8192, 9261, 10648, 12167, 13824, 16807, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653
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OFFSET
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1,1
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COMMENTS
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Terms of A001597 that are not in A000290.
All terms of this sequence are also in A070265 (odd powers), but omitting those odd powers that are also a square (e.g. 64=4^3=8^2).
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Power.
Eric Weisstein's World of Mathematics, Odd Power.
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FORMULA
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A052409(a(n)) is odd. - Reinhard Zumkeller, Mar 28 2014
Sum_{n>=1} 1/a(n) = 1 - zeta(2) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 0.2295303015... - Amiram Eldar, Dec 21 2020
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MAPLE
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# uses code of A001597
for n from 4 do
if not issqr(n) and isA001597(n) then
printf("%d, \n", n);
end if;
end do: # R. J. Mathar, Jan 13 2021
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MATHEMATICA
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nn = 50653; Select[Union[Flatten[Table[n^i, {i, Prime[Range[2, PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]], ! IntegerQ[Sqrt[#]] &] (* T. D. Noe, Apr 19 2011 *)
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PROG
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(Haskell)
import Data.Map (singleton, findMin, deleteMin, insert)
a097054 n = a097054_list !! (n-1)
a097054_list = f 9 (3, 2) (singleton 4 (2, 2)) where
f zz (bz, be) m
| xx < zz && even be =
f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
| xx < zz = xx :
f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
| xx > zz = f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)
| otherwise = f (zz + 2 * bz + 1) (bz + 1, 2) m
where (xx, (bx, be)) = findMin m
-- Reinhard Zumkeller, Mar 28 2014
(PARI) is(n)=ispower(n)%2 \\ Charles R Greathouse IV, Aug 28 2016
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CROSSREFS
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Cf. A001597 (perfect powers), A000290 (the squares), A008683, A070265 (odd powers), A097055, A097056, A239870, A239728, A093771.
Sequence in context: A116002 A339595 A335988 * A304291 A056824 A297868
Adjacent sequences: A097051 A097052 A097053 * A097055 A097056 A097057
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KEYWORD
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nonn
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AUTHOR
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Hugo Pfoertner, Jul 21 2004
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STATUS
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approved
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