A072777 Powers of squarefree numbers that are not squarefree.

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 169, 196, 216, 225, 243, 256, 289, 343, 361, 441, 484, 512, 529, 625, 676, 729, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849

Offset

1,1

Comments

For all n exists k: a(n) = A072774(k) and A072776(k) > 1.

Links

Stanislav Sykora and Reinhard Zumkeller, Table of n, a(n) for n = 1..20000 (first 10000 terms from Reinhard Zumkeller)

Formula

Sum_{n>=1} 1/a(n) = Sum_{n>=2} mu(n)^2/(n*(n-1)) = Sum_{n>=2} (zeta(n)/zeta(2*n) - 1) = 0.8486338679... (A368250). - Amiram Eldar, Jul 22 2020

Example

The number 144 = 12^2 is not a member because 12 is not squarefree.
64 = 2^6 and 49 = 7^2 are members because, though not squarefree, they are powers of the squarefree numbers 2 and 7, respectively. Note that 64 is included even though it is also a square of a nonsquarefree number. - Stanislav Sykora, Jul 11 2014

Mathematica

Select[Range[2000], Length[u = Union[FactorInteger[#][[All, 2]]]] == 1 && u[[1]] > 1 &] (* Jean-François Alcover, Mar 27 2013 *)

Prog

(Haskell)
import Data.Map (singleton, findMin, deleteMin, insert)
a072777 n = a072777_list !! (n-1)
a072777_list = f 9 (drop 2 a005117_list) (singleton 4 (2, 2)) where
   f vv vs'@(v:ws@(w:_)) m
    | xx < vv = xx : f vv vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    | xx > vv = vv : f (w*w) ws (insert (v^3) (v, 3) m)
    where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014
(PARI) BelongsToA(n) = {my(f, k, e); if(n == 1, return(0));
f = factor(n); e = f[1, 2]; if(e == 1, return(0));
for(k = 2, #f[, 2], if(f[k, 2] != e, return(0))); return(1); }
Ntest(nmax, test) = {my(k = 1, n = 0, v); v = vector(nmax); while(1, n++; if(test(n), v[k] = n; k++; if(k > nmax, break)); ); return(v); }
a = Ntest(20000, BelongsToA) \\ Note: not very efficient. - Stanislav Sykora, Jul 11 2014
(PARI) is(n)=ispower(n, , &n) && issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

Crossrefs

Cf. A013929, A368250.

Cf. A005117, subsequence of A001597 and A072774.

Sequence in context: A157985 A001597 A359493 * A076292 A090516 A090515

Adjacent sequences: A072774 A072775 A072776 * A072778 A072779 A072780

Keyword

nonn,nice

Author

Reinhard Zumkeller, Jul 10 2002

Status

approved