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a(n) = 1 + Sum_{k=1..2*n} sign((sign(n+Sum_{j=2..k}-|A309229(n,j)|)+1)).
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%I #27 Jul 22 2020 13:23:05

%S 3,5,5,5,6,7,9,10,7,11,8,13,8,15,12,11,10,11,11,13,13,14,13,14,13,15,

%T 14,15,13,19,17,17,17,19,16,19,15,14,17,17,15,22,17,23,20,19,17,19,17,

%U 19,19,21,19,21,19,21,21,21,21,23,22,22,22,19,21,23,23,23

%N a(n) = 1 + Sum_{k=1..2*n} sign((sign(n+Sum_{j=2..k}-|A309229(n,j)|)+1)).

%C a(n) appears to be asymptotic to sqrt(8*n). Taken from the comment by Lekraj Beedassy in A003418: "An assertion equivalent to the Riemann hypothesis is:

%C | Sum_{k>=1} (A309229(n, k)/k - 1/k) - n | < sqrt(n) * log(n)^2."

%F a(n) = 1 + Sum_{k=1..2*n} sign((sign(n+Sum_{j=2..k}-|A309229(n,j)|)+1)).

%t nn = 68; f[n_] := n; h[n_] := DivisorSum[n, MoebiusMu[#] # &]; A = Accumulate[Table[Table[h[GCD[n, k]], {k, 1, nn}], {n, 1, nn}]]; B = -Abs[A]; B[[All, 1]] = Table[f[n], {n, 1, nn}]; b = 1 + Total[Sign[1 + Sign[Accumulate[Transpose[B]]]]]

%Y Cf. A309229.

%K nonn

%O 1,1

%A _Mats Granvik_, Jun 09 2020