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Number of divisors of the squarefree numbers: tau(A005117(n)).
10

%I #41 Aug 13 2024 01:56:06

%S 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,

%T 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,

%U 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

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

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

%H Reinhard Zumkeller, <a href="/A072048/b072048.txt">Table of n, a(n) for n = 1..10000</a>

%H B. Gordon and K. Rogers, <a href="https://doi.org/10.4153/CJM-1964-015-x">Sums of the divisor function</a>, Canadian Journal of Mathematics, Vol. 16 (1964), pp. 151-158.

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

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

%F 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

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

%t DivisorSigma[0, Select[Range[200], SquareFreeQ]] (* _Amiram Eldar_, Oct 29 2022 *)

%o (Haskell)

%o a072048 = (2 ^) . a072047 -- _Reinhard Zumkeller_, Dec 13 2015

%o (Python)

%o from math import isqrt

%o from sympy import mobius, divisor_count

%o def A072048(n):

%o def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o return divisor_count(m) # _Chai Wah Wu_, Aug 12 2024

%Y Cf. A000005, A001620, A062822, A065473.

%Y Cf. A000079, A001221, A005117, A072047.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jun 09 2002