OFFSET
1,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*(zeta(3*s)/zeta(6*s) - 1), s > 1/2. - Amiram Eldar, Apr 06 2023
MATHEMATICA
Powerful[n_Integer] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Select[Range[10000], Powerful[ # ] && !IntegerQ[Sqrt[ # ]]&] - T. D. Noe, May 03 2006
PROG
(PARI) omnipnotsq(n, m)= local(a, x, j, fl=0); for(x=1, n, a=factor(x); for(j=1, omega(x), if(a[j, 2]>= m, fl=1, fl=0; break); ); if(fl&issquare(x)==0, print1(x", ")) )
(PARI) is(n)=ispowerful(n) && !issquare(n) \\ Charles R Greathouse IV, Oct 19 2015
(Python)
from math import isqrt
from sympy import integer_nthroot, mobius
def A102834(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
def f(x):
j = isqrt(x)
c, l = n+x+j, 0
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, n, n) # Chai Wah Wu, Sep 13 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Feb 27 2005
STATUS
approved