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A376092
10^n-th powerful number.
2
1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, 212104218976916644, 21190268970925690248, 2118092209873957381248, 211765852717674823741924, 21174572668805230623003225, 2117363857447354911021280900
OFFSET
0,2
LINKS
FORMULA
a(n) = A001694(10^n).
Limit_{n->oo} a(n)/10^(2n) = (zeta(3)/zeta(3/2))^2 = 0.21172829478335...
PROG
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A376092(n):
def squarefreepi(n):
return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
m = 10**n
def f(x):
c, l = m+x, 0
j = isqrt(x)
while j>1:
k2 = integer_nthroot(x//j**2, 3)[0]+1
w = squarefreepi(k2-1)
c -= j*(w-l)
l, j = w, isqrt(x//k2**3)
c -= squarefreepi(integer_nthroot(x, 3)[0])-l
return c
return bisection(f, m, m)
CROSSREFS
Sequence in context: A307811 A123841 A014773 * A132539 A248167 A218590
KEYWORD
nonn,more
AUTHOR
Chai Wah Wu, Sep 09 2024
STATUS
approved