A357820 Numerators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

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

Offset

1,2

Links

Table of n, a(n) for n=1..46.

Olivier Bordellès and Benoit Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Formula

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).

Example

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, ...

Mathematica

psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Numerator[Accumulate[1/Array[(-1)^(# + 1)*psi[#] &, 50]]]

Prog

(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

Crossrefs

Cf. A001615, A173290, A357821 (denominators).

Cf. A001620, A065463, A335707.

Similar sequence: A211177.

Sequence in context: A333861 A338845 A121713 * A134242 A087712 A180702

Adjacent sequences: A357817 A357818 A357819 * A357821 A357822 A357823

Keyword

nonn,frac

Author

Amiram Eldar, Oct 14 2022

Status

approved