Amiram Eldar, <a href="/A370897/b370897_1.txt">Table of n, a(n) for n = 1..10000</a>

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Amiram Eldar, <a href="/A370897/b370897_1.txt">Table of n, a(n) for n = 1..10000</a>

Partial alternating sums of the number of abelian groups function sequence (A000688).

allocated for Amiram EldarPartial alternating sums of the number of abelian groups function (A000688).

1, 0, 1, -1, 0, -1, 0, -3, -1, -2, -1, -3, -2, -3, -2, -7, -6, -8, -7, -9, -8, -9, -8, -11, -9, -10, -7, -9, -8, -9, -8, -15, -14, -15, -14, -18, -17, -18, -17, -20, -19, -20, -19, -21, -19, -20, -19, -24, -22, -24, -23, -25, -24, -27, -26, -29, -28, -29, -28

1,8

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.

a(n) = Sum_{k=1..n} (-1)^(k+1) * A000688(k).

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).

f[n_] := Times @@ (PartitionsP[Last[#]] & /@ FactorInteger[n]); f[1] = 1; Accumulate[Array[(-1)^(#+1) * f[#] &, 100]]

(PARI) f(n) = vecprod(apply(numbpart, factor(n)[, 2]));

lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * f(k); print1(s, ", "))};

Cf. A000688, A063966.

Cf. A021002, A048651, A084892, A084893.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

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Amiram Eldar, Mar 05 2024

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