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%I #8 Mar 05 2024 11:52:05
%S 1,0,1,-1,0,-1,0,-3,-1,-2,-1,-3,-2,-3,-2,-7,-6,-8,-7,-9,-8,-9,-8,-11,
%T -9,-10,-7,-9,-8,-9,-8,-15,-14,-15,-14,-18,-17,-18,-17,-20,-19,-20,
%U -19,-21,-19,-20,-19,-24,-22,-24,-23,-25,-24,-27,-26,-29,-28,-29,-28
%N Partial alternating sums of the number of abelian groups sequence (A000688).
%H Amiram Eldar, <a href="/A370897/b370897.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) * A000688(k).
%F a(n) = k_1 * A021002 * n + k_2 * A084892 * n^(1/2) + k_3 * A084893 * n^(1/3) + O(n^(1/4 + eps)), where eps > 0 is arbitrarily small, k_j = -1 + 2 * Product_{i>=1} (1 - 1/2^(i/j)), k_1 = 2*A048651 - 1 = -0.422423809826..., k_2 = -0.924973966404..., and k_3 = -0.991478298912... (Tóth, 2017).
%t f[n_] := Times @@ (PartitionsP[Last[#]] & /@ FactorInteger[n]); f[1] = 1; Accumulate[Array[(-1)^(#+1) * f[#] &, 100]]
%o (PARI) f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
%o lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * f(k); print1(s, ", "))};
%Y Cf. A000688, A063966.
%Y Cf. A021002, A048651, A084892, A084893.
%Y Similar sequences: A068762, A068773, A307704, A357817, A362028.
%K sign,easy
%O 1,8
%A _Amiram Eldar_, Mar 05 2024
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