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%I #39 Jun 13 2023 20:22:02
%S -1,3,12,-52,-27,1269,1318,5414,4685,14685,14806,3000790,3000959,
%T 3039375,3090000,2041424,2041713,36053937,36054298,100054298,
%U 100248779,100483035,100483564,110175797740,110175782115,110176239091,110176770532,110658660836
%N Partial sums of A217854.
%C If there is some n > 47 such that a(n) < 0, then there is some k^2 > 47 such that a(k^2) < 0.
%C If n > 1 is a square number, then a(n) = a(n-1) - n^tau(n).
%C If n > 1 is a nonsquare number, then a(n) = a(n-1) + n^tau(n).
%C If n > 1 is a prime, then a(n) = a(n-1) + n^2.
%H Simon Jensen, <a href="/A224914/b224914.txt">Table of n, a(n) for n = 1..10000</a>
%H Simon Jensen, <a href="https://www.simonjensen.com/pdf/On_an_extended_divisor_product_summatory_function.pdf">On an extended divisor product summatory function</a>
%F a(n) = Sum_{i=1..n} (-i)^tau(i) = Sum_{i=1..n} (-i)^A000005(i) = Sum_{i=1..n} A217854(i).
%e a(4) = a(1) + a(2) + a(3) + (-4)^tau(4) = (-1) + 3 + 12 + (-64) = -52.
%t Accumulate@ Table[(-n)^DivisorSigma[0, n], {n, 28}] (* _Michael De Vlieger_, Mar 18 2016 *)
%o (PARI) a(n) = sum(k=1, n, (-k)^numdiv(k)); \\ _Michel Marcus_, Mar 18 2016
%K sign
%O 1,2
%A _Simon Jensen_, Apr 19 2013
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