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Partial alternating sums of the sum of unitary divisors function (A034448).
6

%I #7 Mar 05 2024 11:51:52

%S 1,-2,2,-3,3,-9,-1,-10,0,-18,-6,-26,-12,-36,-12,-29,-11,-41,-21,-51,

%T -19,-55,-31,-67,-41,-83,-55,-95,-65,-137,-105,-138,-90,-144,-96,-146,

%U -108,-168,-112,-166,-124,-220,-176,-236,-176,-248,-200,-268,-218,-296,-224,-294,-240,-324,-252,-324,-244,-334,-274,-394

%N Partial alternating sums of the sum of unitary divisors function (A034448).

%H Amiram Eldar, <a href="/A370898/b370898.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) * A034448(k).

%F a(n) = -c * n^2 + O(n * log(n)^(5/3)), where c = Pi^2/(84*zeta(3)) = 0.0977451984014... (Tóth, 2017).

%t usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Accumulate[Array[(-1)^(# + 1) * usigma[#] &, 100]]

%o (PARI) usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}

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

%Y Cf. A002117, A034448, A064609.

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

%K sign,easy

%O 1,2

%A _Amiram Eldar_, Mar 05 2024