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A072777 Powers of squarefree numbers that are not squarefree. 8
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 (list; graph; refs; listen; history; text; internal format)
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) = 0.8486332567... - 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-Fran├žois 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.

Cf. A005117, subsequence of A001597 and A072774.

Sequence in context: A317102 A157985 A001597 * A076292 A090516 A090515

Adjacent sequences:  A072774 A072775 A072776 * A072778 A072779 A072780

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Jul 10 2002

STATUS

approved

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Last modified September 19 15:09 EDT 2020. Contains 337178 sequences. (Running on oeis4.)