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A072777 Powers of squarefree numbers that are not squarefree.

%I A072777 #57 Aug 19 2024 11:38:51
%S A072777 4,8,9,16,25,27,32,36,49,64,81,100,121,125,128,169,196,216,225,243,
%T A072777 256,289,343,361,441,484,512,529,625,676,729,841,900,961,1000,1024,
%U A072777 1089,1156,1225,1296,1331,1369,1444,1521,1681,1764,1849
%N A072777 Powers of squarefree numbers that are not squarefree.
%C A072777 For all n exists k: a(n) = A072774(k) and A072776(k) > 1.
%C A072777 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
%H A072777 Stanislav Sykora and Reinhard Zumkeller, <a href="/A072777/b072777.txt">Table of n, a(n) for n = 1..20000</a> (first 10000 terms from Reinhard Zumkeller)
%F A072777 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
%e A072777 The number 144 = 12^2 is not a member because 12 is not squarefree.
%e A072777 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
%t A072777 Select[Range[2000], Length[u = Union[FactorInteger[#][[All, 2]]]] == 1 && u[[1]] > 1 &] (* _Jean-François Alcover_, Mar 27 2013 *)
%o A072777 (Haskell)
%o A072777 import Data.Map (singleton, findMin, deleteMin, insert)
%o A072777 a072777 n = a072777_list !! (n-1)
%o A072777 a072777_list = f 9 (drop 2 a005117_list) (singleton 4 (2, 2)) where
%o A072777    f vv vs'@(v:ws@(w:_)) m
%o A072777     | xx < vv = xx : f vv vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
%o A072777     | xx > vv = vv : f (w*w) ws (insert (v^3) (v, 3) m)
%o A072777     where (xx, (bx, ex)) = findMin m
%o A072777 -- _Reinhard Zumkeller_, Apr 06 2014
%o A072777 (PARI) BelongsToA(n) = {my(f, k, e); if(n == 1, return(0));
%o A072777 f = factor(n); e = f[1, 2]; if(e == 1, return(0));
%o A072777 for(k = 2, #f[, 2], if(f[k, 2] != e, return(0))); return(1);}
%o A072777 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); }
%o A072777 a = Ntest(20000, BelongsToA) \\ Note: not very efficient. - _Stanislav Sykora_, Jul 11 2014
%o A072777 (PARI) is(n)=ispower(n,,&n) && issquarefree(n) \\ _Charles R Greathouse IV_, Oct 16 2015
%o A072777 (Python)
%o A072777 from math import isqrt
%o A072777 from sympy import mobius, integer_nthroot
%o A072777 def A072777(n):
%o A072777     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
%o A072777     def f(x): return n-1+x-sum(g(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))
%o A072777     kmin, kmax = 1,2
%o A072777     while f(kmax) >= kmax:
%o A072777         kmax <<= 1
%o A072777     while True:
%o A072777         kmid = kmax+kmin>>1
%o A072777         if f(kmid) < kmid:
%o A072777             kmax = kmid
%o A072777         else:
%o A072777             kmin = kmid
%o A072777         if kmax-kmin <= 1:
%o A072777             break
%o A072777     return kmax # _Chai Wah Wu_, Aug 19 2024
%Y A072777 Cf. A013929, A368250.
%Y A072777 Cf. A005117, subsequence of A001597 and A072774.
%Y A072777 Cf. A007947.
%K A072777 nonn,nice
%O A072777 1,1
%A A072777 _Reinhard Zumkeller_, Jul 10 2002