OFFSET
1,1
COMMENTS
Former name: Numbers k such that k/(largest power of squarefree kernel of k) is larger than 1.
Complement of A072774 (powers of squarefree numbers).
Also numbers k = p(1)^alpha(1)* ... * p(r)^alpha(r) such that RootMeanSquare(alpha(1), ..., alpha(r)) is not an integer. - Ctibor O. Zizka, Sep 19 2008
LINKS
Donald Alan Morrison, Table of n, a(n) for n = 1..10000
Donald Alan Morrison, Sage program
FORMULA
EXAMPLE
440 is in the sequence because 440 = 2^3*5*11 and it has two distinct exponents, 3 and 1.
PROG
(PARI) is(n)=#Set(factor(n)[, 2])>1 \\ Charles R Greathouse IV, Sep 18 2015
(Python)
from sympy import factorint
def ok(n): return len(set(factorint(n).values())) > 1
print([k for k in range(190) if ok(k)]) # Michael S. Branicky, Sep 01 2022
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A059404(n):
def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
def f(x): return n+1-(y:=x.bit_length())+sum(g(integer_nthroot(x, k)[0]) for k in range(1, y))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
return kmax # Chai Wah Wu, Aug 19 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Jul 18 2001
STATUS
approved