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%I #7 Mar 05 2024 11:52:10
%S 1,-1,2,0,5,-1,6,4,7,-3,8,2,15,1,16,14,31,25,44,34,55,33,56,50,55,29,
%T 32,18,47,17,48,46,79,45,80,74,111,73,112,102,143,101,144,122,137,91,
%U 138,132,139,129,180,154,207,201,256,242,299,241,300,270,331,269
%N Partial alternating sums of the squarefree kernel function (A007947).
%H Amiram Eldar, <a href="/A370896/b370896.txt">Table of n, a(n) for n = 1..10000</a>
%H László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
%F a(n) = Sum_{k=1..n} (-1)^(k+1) * A007947(k).
%F a(n) = c * n^2 + O(R(n)), where c = A065463 / 10 = 0.07044422..., R(n) = x^(3/2)*exp(-c_1*log(n)^(3/5)/log(log(n))^(1/5)) unconditionally, or x^(7/5)*exp(c_2*log(n)/log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).
%t rad[n_] := Times @@ (First[#]& /@ FactorInteger[n]); Accumulate[Array[(-1)^(#+1) * rad[#] &, 100]]
%o (PARI) rad(n) = vecprod(factor(n)[, 1]);
%o lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * rad(k); print1(s, ", "))};
%Y Cf. A007947, A065463, A073355.
%Y Similar sequences: A068762, A068773, A307704, A357817, A362028.
%K sign,easy
%O 1,3
%A _Amiram Eldar_, Mar 05 2024