OFFSET
1,2
COMMENTS
Perfect-powers (A001597) are numbers with a proper integer root.
MATHEMATICA
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All, 2]]>1;
per=Select[Range[1000], perpowQ];
per-NestList[NestWhile[#+1&, #+1, !PrimePowerQ[#]&]&, 1, Length[per]-1]
PROG
(Python)
from sympy import mobius, primepi, integer_nthroot
def A377043(n):
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): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
def g(x): return int(n-1+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length())))
return bisection(f, n, n)-bisection(g, n, n) # Chai Wah Wu, Oct 27 2024
CROSSREFS
Excluding 1 from the powers of primes gives A377044.
A000015 gives the least prime-power >= n.
A031218 gives the greatest prime-power <= n.
A025475 lists numbers that are both a perfect-power and a prime-power.
A080101 counts prime-powers between primes (exclusive).
A106543 lists numbers that are neither a perfect-power nor a prime-power.
A131605 lists perfect-powers that are not prime-powers.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 25 2024
STATUS
approved