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A376092 10^n-th powerful number.

%I A376092 #25 Jul 21 2025 12:38:43
%S A376092 1,49,3136,253472,23002083,2200079025,215523459072,21348015504200,
%T A376092 2125390162618116,212104218976916644,21190268970925690248,
%U A376092 2118092209873957381248,211765852717674823741924,21174572668805230623003225,2117363857447354911021280900
%N A376092 10^n-th powerful number.
%H A376092 Chai Wah Wu, <a href="/A376092/b376092.txt">Table of n, a(n) for n = 0..16</a>
%F A376092 a(n) = A001694(10^n).
%F A376092 Limit_{n->oo} a(n)/10^(2n) = (zeta(3)/zeta(3/2))^2 = 0.21172829478335...
%o A376092 (Python)
%o A376092 from math import isqrt
%o A376092 from sympy import mobius, integer_nthroot
%o A376092 def A376092(n):
%o A376092     def squarefreepi(n):
%o A376092         return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A376092     def bisection(f, kmin=0, kmax=1):
%o A376092         while f(kmax) > kmax: kmax <<= 1
%o A376092         while kmax-kmin > 1:
%o A376092             kmid = kmax+kmin>>1
%o A376092             if f(kmid) <= kmid:
%o A376092                 kmax = kmid
%o A376092             else:
%o A376092                 kmin = kmid
%o A376092         return kmax
%o A376092     m = 10**n
%o A376092     def f(x):
%o A376092         c, l = m+x, 0
%o A376092         j = isqrt(x)
%o A376092         while j>1:
%o A376092             k2 = integer_nthroot(x//j**2,3)[0]+1
%o A376092             w = squarefreepi(k2-1)
%o A376092             c -= j*(w-l)
%o A376092             l, j = w, isqrt(x//k2**3)
%o A376092         c -= squarefreepi(integer_nthroot(x,3)[0])-l
%o A376092         return c
%o A376092     return bisection(f,m,m)
%Y A376092 Cf. A001694, A118896.
%K A376092 nonn
%O A376092 0,2
%A A376092 _Chai Wah Wu_, Sep 09 2024