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A143348 a(n) = -(-1)^n times sum of divisors of n.

%I A143348 #23 Oct 28 2023 04:05:01
%S A143348 1,-3,4,-7,6,-12,8,-15,13,-18,12,-28,14,-24,24,-31,18,-39,20,-42,32,
%T A143348 -36,24,-60,31,-42,40,-56,30,-72,32,-63,48,-54,48,-91,38,-60,56,-90,
%U A143348 42,-96,44,-84,78,-72,48,-124,57,-93,72,-98,54,-120,72,-120,80,-90,60,-168,62,-96,104,-127,84,-144,68,-126,96
%N A143348 a(n) = -(-1)^n times sum of divisors of n.
%H A143348 Indranil Ghosh, <a href="/A143348/b143348.txt">Table of n, a(n) for n = 1..10000</a>
%H A143348 László Tóth, <a href="https://arxiv.org/abs/1608.00795">Alternating sums concerning multiplicative arithmetic functions</a>, arXiv preprint arXiv:1608.00795 [math.NT], 2016.
%F A143348 a(n) is multiplicative with a(2^e) = 1 - 2^(e+1) if e > 0, a(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2.
%F A143348 G.f.: Sum_{k>0} k * -(-x)^k / (1 - (-x)^k) = Sum_{k>0} -(-x)^k / (1 - (-x)^k)^2.
%F A143348 Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 6 / 2^s + 4 / 4^s).
%F A143348 Sum_{k=1..n} a(k) ~ -(Pi^2/48) * n^2. - _Amiram Eldar_, Oct 28 2023
%e A143348 q - 3*q^2 + 4*q^3 - 7*q^4 + 6*q^5 - 12*q^6 + 8*q^7 - 15*q^8 + 13*q^9 + ...
%t A143348 Table[-(-1)^n*DivisorSigma[1, n], {n, 69}] (* _Michael De Vlieger_, Aug 19 2017 *)
%o A143348 (PARI) {a(n) = if( n<1, 0, -(-1)^n * sigma(n))}
%Y A143348 -(-1)^n * A000203(n) = a(n). A143337(n) = 24 * a(n) unless n=0.
%Y A143348 Cf. A245058 (-Pi^2/48).
%K A143348 sign,easy,mult
%O A143348 1,2
%A A143348 _Michael Somos_, Aug 09 2008