%I #13 Sep 11 2024 00:34:43
%S 1,0,0,1,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0,0,1,1,0,0,1,0,1,1,0,0,0,
%T 1,0,0,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,
%U 0,0,1,0,0,0,1,0,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0,0,1,0,0,1,1,1,0,0,1,1,0,1
%N a(n) is the parity of the n-th powerful number.
%H Amiram Eldar, <a href="/A373549/b373549.txt">Table of n, a(n) for n = 1..10000</a>
%H Teerapat Srichan, <a href="http://www.math.nthu.edu.tw/~amen/2020/AMEN-200219.pdf">The odd/even dichotomy for the set of square-full numbers</a>, Applied Mathematics E-Notes, Vol. 20 (2020), pp. 528-531.
%H Wikipedia, <a href="https://oeis.org/A363176/b363176.txt">Powerful number</a>.
%F a(n) = A001694(n) mod 2 = A000035(A001694(n)).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (2 - sqrt(2))/(3 - sqrt(2)) = 0.3693980625... .
%t Mod[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;;, 2]]] > 1 &], 2]
%o (PARI) lista(kmax) = for(k = 1, kmax, if(ispowerful(k), print1(k % 2, ", ")));
%o (Python)
%o from math import isqrt
%o from sympy import mobius, integer_nthroot
%o def A373549(n):
%o def squarefreepi(n):
%o return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o def bisection(f, kmin=0, kmax=1):
%o while f(kmax) > kmax: kmax <<= 1
%o while kmax-kmin > 1:
%o kmid = kmax+kmin>>1
%o if f(kmid) <= kmid:
%o kmax = kmid
%o else:
%o kmin = kmid
%o return kmax
%o def f(x):
%o c, l = n+x, 0
%o j = isqrt(x)
%o while j>1:
%o k2 = integer_nthroot(x//j**2, 3)[0]+1
%o w = squarefreepi(k2-1)
%o c -= j*(w-l)
%o l, j = w, isqrt(x//k2**3)
%o c -= squarefreepi(integer_nthroot(x, 3)[0])-l
%o return c
%o return bisection(f,n,n)&1 # _Chai Wah Wu_, Sep 10 2024
%Y Characteristic function of A363189.
%Y Cf. A000035, A001694, A062739, A335851, A363191, A363192, A373550.
%K nonn,easy
%O 1
%A _Amiram Eldar_, Jun 09 2024