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%I #31 Sep 08 2022 08:45:05
%S 0,1,3,6,7,12,18,25,27,28,38,49,52,65,79,94,95,112,114,133,138,159,
%T 181,204,210,211,237,240,247,276,306,337,339,372,406,441,442,479,517,
%U 556,566,607,649,692,703,708,754,801,804,805,807,858,871,924,930,985,999
%N a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.
%C Sum_{k=1..n, k squarefree} (1/k) = Sum{k=1..n} (mu(k)^2/k) = (1/zeta(2))*(log(n) + gamma - 2*zeta'(2)/zeta(2)) + O(1/sqrt(n)). (Suryanarayana)
%D D. Suryanarayana, Asymptotic formula for sum_{n <= x} mu^{2}(n)/n, Indian J. Math. 9 (1967/1968) 543-545.
%H Amiram Eldar, <a href="/A069891/b069891.txt">Table of n, a(n) for n = 0..10000</a>
%H Rafael Jakimczuk, <a href="https://dx.doi.org/10.12988/imf.2017.7113">Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts</a>, International Mathematical Forum, Vol. 12, 2017, no. 7, pp. 331-338.
%F a(n) = Sum_{d=1..floor(sqrt(n))} f(d)*binomial(floor(n/d^2)+1, 2) where f(d)=A046970(d) is the product of 1-p^2 over all prime divisors p of d.
%F a(n) is asymptotic to r*n^2, where r = Pi^2/30 = 0.3289868...
%t a[n_] := Sum[If[d == 1, 1, Times@@(1-#1[[1]]^2&) /@ FactorInteger[d]] * Binomial[Floor[n/d^2]+1, 2], {d, 1, Floor[Sqrt[n]]}]; Array[a, 100, 0] (* corrected by _Amiram Eldar_, Apr 02 2020 *)
%o (Magma) [0] cat [&+[Squarefree(k):k in [1..n]]:n in [1..60]]; // _Marius A. Burtea_, Dec 19 2019
%o (PARI) a(n) = sum(k=1, n, core(k)); \\ _Michel Marcus_, Dec 19 2019
%Y Cf. A007913, A046970, A069087.
%K nonn
%O 0,3
%A _Dean Hickerson_, Apr 09 2002
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