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Denominators of first sequence: A003418, which are symmetric sequences of greatest divisors <= ii. The denominators of the differences are these greatest divisors. See here and here. Edit: I missed that ii=9 is 840 instead of 2520 in the first sequence and is a zero instead of 9 in the second. So, the symmetry stops at 9. I'll leave this answer in case ...

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my stab at it, unfortunately only a marginal improvement in time: ii=13; Clear[a, b]; b = FoldList[Times, 1, Table[ Exp[MangoldtLambda[n]], {n, 2, ii}]]; a = Prepend[Table[ Limit[ Zeta[s] Total[Exp[Divisors[n]]^(s - 1) MoebiusMu[Divisors[n]]], s -> 1], {n, 2, ii}], 1] ; Monitor[aa = Prepend[Table[ ...

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I realized now that I included unnecessary many terms of the Dirichlet inverse of the Euler totient. Therefore a better program is: ii = 13 aa = Range[ii]*0; Monitor[Do[ Clear[A, a, b, n, k]; b = Table[Product[Exp[MangoldtLambda[n]], {n, 1, k}], {k, 1, nn}]; a = Table[ If[n == 1, 1, Limit[Zeta[s] Total[ Exp[Divisors[n]]^(s - ...

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