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Partial alternating sums of the sum of the bi-unitary divisors function (A188999).
5

%I #7 Mar 05 2024 11:50:46

%S 1,-2,2,-3,3,-9,-1,-16,-6,-24,-12,-32,-18,-42,-18,-45,-27,-57,-37,-67,

%T -35,-71,-47,-107,-81,-123,-83,-123,-93,-165,-133,-196,-148,-202,-154,

%U -204,-166,-226,-170,-260,-218,-314,-270,-330,-270,-342,-294,-402,-352,-430

%N Partial alternating sums of the sum of the bi-unitary divisors function (A188999).

%H Amiram Eldar, <a href="/A370904/b370904.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) * A188999(k).

%F a(n) = -(11/53) * c * n^2 + O(n * log(n)^3), where c = A307160 (Tóth, 2017).

%t f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(# + 1) * bsigma[#] &, 100]]

%o (PARI) bsigma(n) = {my(f = factor(n)); prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)));}

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

%Y Cf. A188999, A307159, A307160.

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

%K sign,easy

%O 1,2

%A _Amiram Eldar_, Mar 05 2024