A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).

1, 2, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4, 8, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 2, 8, 4, 2

Offset: 1

Author

Reinhard Zumkeller, Jun 09 2002

Keywords

• nonn

Comments

Also the number of cubefree numbers with the same squarefree kernel as the n-th squarefree number, see A073245.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• B. Gordon and K. Rogers, Sums of the divisor function, Canadian Journal of Mathematics, Vol. 16 (1964), pp. 151-158.

Crossrefs

Cf. A000005, A001620, A062822, A065473.

Cf. A000079, A001221, A005117, A072047.

Programs

Haskell

a072048 = (2 ^) . a072047  -- Reinhard Zumkeller, Dec 13 2015

Maple

A072048:=n->`if`(numtheory[issqrfree](n) = true, numtheory[tau](n), NULL); seq(A072048(k), k=1..100); # Wesley Ivan Hurt, Oct 13 2013

Mathematica

DivisorSigma[0, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 29 2022 *)

Python

from math import isqrt
from sympy import mobius, divisor_count
def A072048(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_count(m) # Chai Wah Wu, Aug 12 2024

Formula

a(n) = A000005(A005117(n)).

a(n) = 2^A072047(n) = 2^A001221(A005117(n)).

Sum_{k=1..n} a(k) ~ A * n * log(n) + B * n + O(n^(1/2+eps)), where A = A065473, B = A * ((2*gamma-1) + 6 * Sum_{p prime} (p-1)*log(p)/(p^2*(p+2)) = 0.236184..., and gamma = A001620 (Gordon and Rogers, 1964). - Amiram Eldar, Oct 29 2022