A076400 Number of divisors of n-th perfect power.

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

Cf. A000005, A001597, A076398, A076399, A076401.

Sequence in context: A201420 A244055 A090739 * A363194 A121889 A332026

Adjacent sequences: A076397 A076398 A076399 * A076401 A076402 A076403

Keyword

nonn

Author

Reinhard Zumkeller, Oct 09 2002

Status

approved