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

Author

Reinhard Zumkeller, Jul 10 2002

Keywords

• nonn
• nice

Comments

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

Numbers k such that every prime in the prime factorization of k is raised to the same power > 1; k is a term iff k/A007947(k)^m = 1 for some m > 1. - David James Sycamore, Jun 12 2024

Examples

			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
		

Links

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

Crossrefs

Cf. A013929, A368250.

Cf. A005117, subsequence of A001597 and A072774.

Cf. A007947.

Programs

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

Mathematica

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

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

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A072777(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-1+x-sum(g(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

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