OFFSET
1,1
COMMENTS
Powerful numbers that are not squares of squarefree numbers. - Amiram Eldar, Jun 25 2022
LINKS
Hugo Pfoertner, Table of n, a(n) for n = 1..10000
FORMULA
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 15/Pi^2 = 0.4237786821... . - Amiram Eldar, Jun 25 2022
MATHEMATICA
Select[Range[2500], (m = MinMax[FactorInteger[#][[;; , 2]]])[[1]] > 1 && m[[2]] > 2 &] (* Amiram Eldar, Jun 25 2022 *)
PROG
(PARI) isA001694(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]==1, return(0))); 1 \\ from Charles R Greathouse IV
isA046099(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]>2, return(1))); 0
for (k=1, 2500, if(isA001694(k)&&isA046099(k), print1(k, ", ")))
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A320966(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):
c, l = n+x+squarefreepi(isqrt(x))-squarefreepi(integer_nthroot(x, 3)[0]), 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)
return c+l
return bisection(f, n, n) # Chai Wah Wu, Sep 15 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Oct 25 2018
STATUS
approved