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A097054 Nonsquare perfect powers. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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).
LINKS
Eric Weisstein's World of Mathematics, Perfect Power.
Eric Weisstein's World of Mathematics, Odd Power.
FORMULA
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
MAPLE
# 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
MATHEMATICA
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 *)
PROG
(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
(PARI) list(lim)=my(v=List()); forprime(e=3, logint(lim\=1, 2), for(b=2, sqrtnint(lim, e), if(!issquare(b), listput(v
, b^e)))); Set(v) \\ Charles R Greathouse IV, Jan 09 2023
CROSSREFS
Cf. A001597 (perfect powers), A000290 (the squares), A008683, A070265 (odd powers), A097055, A097056, A239870, A239728, A093771.
Sequence in context: A116002 A339595 A335988 * A370788 A304291 A056824
KEYWORD
nonn,easy
AUTHOR
Hugo Pfoertner, Jul 21 2004
STATUS
approved

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)