OFFSET
1,1
COMMENTS
The sequence contains numbers m such that the exponents e are identical for all prime power factors p^e | m. It is clear from this alternate definition that m / K^E = 1 iff E is an integer. - Michael De Vlieger, Jun 24 2022
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
EXAMPLE
Primes, squarefree numbers and perfect powers are here.
From Michael De Vlieger, Jun 24 2022 (Start):
144 cannot be in the sequence, since the exponents of its prime power factors differ. The squarefree kernel of 144 = 2^4 * 3^2 is 2*3 = 6. The largest power of 6 less than 144 is 36. 144/36 = 4, so it is not in the sequence.
216 is in the sequence because 216 = 2^3 * 3^3 is 2*3 = 6. But 216 = 6^3, hence 6^3 / 6^3 = 1. (End)
MATHEMATICA
Select[Range[2, 2^16], Length@ Union@ FactorInteger[#][[All, -1]] == 1 &] Michael De Vlieger, Jun 24 2022
PROG
(PARI) is(n)=ispower(n, , &n); issquarefree(n) && n>1 \\ Charles R Greathouse IV, Sep 18 2015
(PARI) is(n)=#Set(factor(n)[, 2])==1 \\ Charles R Greathouse IV, Sep 18 2015
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A062770(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-2+x+(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
AUTHOR
Labos Elemer, Jul 18 2001
EXTENSIONS
Offset corrected by Charles R Greathouse IV, Sep 18 2015
STATUS
approved