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%I #27 Feb 22 2024 20:02:15
%S 0,1,-1,3,-1,1,-1,5,-3,1,-1,4,-1,1,-3,8,-1,0,-1,4,-3,1,-1,7,-3,1,-5,4,
%T -1,1,-1,10,-3,1,-3,5,-1,1,-3,7,-1,1,-1,4,-6,1,-1,11,-3,0,-3,4,-1,-1,
%U -3,7,-3,1,-1,6,-1,1,-6,14,-3,1,-1,4,-3,1,-1,9,-1,1,-6,4,-3,1
%N Alternating sum of the k-adic valuations (ruler functions) of n.
%H Alois P. Heinz, <a href="/A369180/b369180.txt">Table of n, a(n) for n = 1..20000</a>
%F a(n) = Sum_{k=2..n} (-1)^k * valuation(n,k).
%F a(n) = A007814(n) - A007949(n) + A235127(n) - (...).
%F G.f.: Sum_{k>=2, j>=1} (-1)^k x^(k^j)/(1-x^(k^j)).
%F Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n) = log(2).
%F Dirichlet g.f.: zeta(s) * Sum_{k>=1} (1 - eta(ks)).
%F Sum_{n>=1} a(n)/n^2 = Pi^2/24.
%p a:= n-> add((-1)^i*padic[ordp](n, i), i=2..n):
%p seq(a(n), n=1..78); # _Alois P. Heinz_, Jan 15 2024
%t z = 70; Sum[(-1)^k IntegerExponent[Range[z], k], {k, 2, z}]
%o (PARI) a(n) = sum(k=2, n, (-1)^k * valuation(n,k)); \\ _Michel Marcus_, Jan 18 2024
%Y Cf. A007814, A007949, A222171, A235127.
%Y Cf. A309891.
%K sign
%O 1,4
%A _Friedjof Tellkamp_, Jan 15 2024
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