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%I #36 Oct 29 2022 04:51:19
%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
%Y Cf. A000005, A001620, A062822, A065473.
%Y Cf. A000079, A001221, A005117, A072047.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Jun 09 2002
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