cp's OEIS Frontend

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

%I A072048 #41 Aug 13 2024 01:56:06
%S A072048 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 A072048 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 A072048 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 A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).
%C A072048 Also the number of cubefree numbers with the same squarefree kernel as the n-th squarefree number, see A073245.
%H A072048 Reinhard Zumkeller, <a href="/A072048/b072048.txt">Table of n, a(n) for n = 1..10000</a>
%H A072048 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 A072048 a(n) = A000005(A005117(n)).
%F A072048 a(n) = 2^A072047(n) = 2^A001221(A005117(n)).
%F A072048 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 A072048:=n->`if`(numtheory[issqrfree](n) = true, numtheory[tau](n), NULL); seq(A072048(k), k=1..100); # _Wesley Ivan Hurt_, Oct 13 2013
%t A072048 DivisorSigma[0, Select[Range[200], SquareFreeQ]] (* _Amiram Eldar_, Oct 29 2022 *)
%o A072048 (Haskell)
%o A072048 a072048 = (2 ^) . a072047  -- _Reinhard Zumkeller_, Dec 13 2015
%o A072048 (Python)
%o A072048 from math import isqrt
%o A072048 from sympy import mobius, divisor_count
%o A072048 def A072048(n):
%o A072048     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A072048     m, k = n, f(n)
%o A072048     while m != k:
%o A072048         m, k = k, f(k)
%o A072048     return divisor_count(m) # _Chai Wah Wu_, Aug 12 2024
%Y A072048 Cf. A000005, A001620, A062822, A065473.
%Y A072048 Cf. A000079, A001221, A005117, A072047.
%K A072048 nonn
%O A072048 1,2
%A A072048 _Reinhard Zumkeller_, Jun 09 2002