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A097054
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Nonsquare perfect powers.
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12
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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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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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Eric Weisstein's World of Mathematics, Odd Power.
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FORMULA
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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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for n from 4 do
if not issqr(n) and isA001597(n) then
printf("%d, \n", n);
end if;
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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
(PARI) list(lim)=my(v=List()); forprime(e=3, logint(lim\=1, 2), for(b=2, sqrtnint(lim, e), if(!issquare(b), listput(v
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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