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%I #8 Oct 28 2020 03:57:15
%S 1,2,0,1,7,5,-1,0,7,13,3,1,15,9,-3,-2,16,23,5,11,23,13,-9,-11,20,34,
%T 14,8,38,26,-4,-3,17,35,-1,6,44,26,-2,4,46,58,16,6,48,26,-20,-22,21,
%U 52,16,30,84,64,4,-2,34,64,6,-6,56,26,-16,-15,69,89,23,41,85,49,-21,-14,60,98,36
%N a(n) = n - 3 * floor(n/3) + 5 * floor(n/5) - 7 * floor(n/7) + ...
%C Partial sums of A050457.
%H Robert Israel, <a href="/A309124/b309124.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: (1/(1 - x)) * Sum_{k>=1} (-1)^(k+1) * (2*k - 1) * x^(2*k-1)/(1 - x^(2*k-1)).
%p f:= proc(n) local r,d;
%p r:= n/2^padic:-ordp(n,2);
%p add((-1)^((d-1)/2)*d, d = numtheory:-divisors(r))
%p end proc:
%p ListTools:-PartialSums(map(f,[$1..100])); # _Robert Israel_, Oct 28 2020
%t Table[Sum[(-1)^(k + 1) (2 k - 1) Floor[n/(2 k - 1)], {k, 1, n}], {n, 1, 75}]
%t nmax = 75; CoefficientList[Series[1/(1 - x) Sum[(-1)^(k + 1) (2 k - 1) x^(2 k - 1)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%Y Cf. A014200, A024919, A050457, A078471.
%K sign
%O 1,2
%A _Ilya Gutkovskiy_, Jul 13 2019
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