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A357820
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Numerators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).
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3
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1, 2, 11, 3, 11, 5, 23, 7, 23, 65, 71, 17, 64, 491, 64, 491, 173, 505, 2651, 2581, 10639, 1151, 3593, 3523, 727, 237, 2189, 2147, 11071, 10931, 5623, 2759, 5623, 16589, 2113, 8347, 162373, 159979, 20318, 160549, 163969, 649891, 7292441, 7204661, 7292441, 7204661
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/psi(k)).
a(n)/A357821(n) ~ (C/5) * (log(n) + gamma + D + 24*log(2)/5) + O(log(n)^(2/3) * log(log(n))^(4/3) / n), where C = Product_{p prime} (1 - 1/(p*(p+1))) (A065463), and D = Sum_{p prime} log(p)/(p^2+p-1) (A335707) (Bordellès and Cloitre, 2013; Tóth, 2017).
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EXAMPLE
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Fractions begin with 1, 2/3, 11/12, 3/4, 11/12, 5/6, 23/24, 7/8, 23/24, 65/72, 71/72, 17/18, ...
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MATHEMATICA
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psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Numerator[Accumulate[1/Array[(-1)^(# + 1)*psi[#] &, 50]]]
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PROG
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(PARI) f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = numerator(sum(k=1, n, (-1)^(k+1)/f(k))); \\ Michel Marcus, Oct 15 2022
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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