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A370904 Partial alternating sums of the sum of the bi-unitary divisors function (A188999).

%I A370904 #10 Apr 26 2025 20:54:57
%S A370904 1,-2,2,-3,3,-9,-1,-16,-6,-24,-12,-32,-18,-42,-18,-45,-27,-57,-37,-67,
%T A370904 -35,-71,-47,-107,-81,-123,-83,-123,-93,-165,-133,-196,-148,-202,-154,
%U A370904 -204,-166,-226,-170,-260,-218,-314,-270,-330,-270,-342,-294,-402,-352,-430
%N A370904 Partial alternating sums of the sum of the bi-unitary divisors function (A188999).
%H A370904 Amiram Eldar, <a href="/A370904/b370904.txt">Table of n, a(n) for n = 1..10000</a>
%H A370904 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 A370904 a(n) = Sum_{k=1..n} (-1)^(k+1) * A188999(k).
%F A370904 a(n) = -(11/53) * c * n^2 + O(n * log(n)^3), where c = A307160 (Tóth, 2017).
%t A370904 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 A370904 (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 A370904 lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * bsigma(k); print1(s, ", "))};
%Y A370904 Cf. A188999, A307159, A307160.
%Y A370904 Similar sequences: A068762, A068773, A307704, A357817, A362028.
%K A370904 sign,easy
%O A370904 1,2
%A A370904 _Amiram Eldar_, Mar 05 2024