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A076400
Number of divisors of n-th perfect power.
10
1, 3, 4, 3, 5, 3, 4, 6, 9, 3, 7, 5, 9, 3, 4, 8, 15, 3, 9, 16, 9, 6, 9, 3, 15, 4, 3, 15, 9, 9, 10, 3, 21, 5, 9, 7, 15, 3, 27, 3, 16, 11, 9, 9, 9, 25, 4, 3, 9, 9, 21, 3, 28, 27, 3, 15, 15, 12, 9, 8, 4, 3, 27, 5, 15, 9, 15, 16, 3, 21, 9, 6, 21, 9, 9, 16, 3, 45, 3, 9, 15, 13, 9, 27, 3, 15, 9, 27, 4
OFFSET
1,2
LINKS
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Divisor Function.
FORMULA
a(n) = A000005(A001597(n)).
MATHEMATICA
DivisorSigma[0, {1}~Join~Select[Range[5000], GCD @@ FactorInteger[#][[All, -1]] > 1 &]] (* Michael De Vlieger, Dec 16 2021 *)
PROG
(Python)
from sympy import mobius, integer_nthroot, divisor_count
def A076400(n):
def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
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 int(divisor_count(kmax)) # Chai Wah Wu, Aug 14 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 09 2002
STATUS
approved