cp's OEIS Frontend

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

%I A357820 #9 Oct 15 2022 07:21:24
%S A357820 1,2,11,3,11,5,23,7,23,65,71,17,64,491,64,491,173,505,2651,2581,10639,
%T A357820 1151,3593,3523,727,237,2189,2147,11071,10931,5623,2759,5623,16589,
%U A357820 2113,8347,162373,159979,20318,160549,163969,649891,7292441,7204661,7292441,7204661
%N A357820 Numerators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).
%H A357820 Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Bordelles/bord14.html">An alternating sum involving the reciprocal of certain multiplicative functions</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3.
%H A357820 László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
%F A357820 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/psi(k)).
%F A357820 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).
%e A357820 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, ...
%t A357820 psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Numerator[Accumulate[1/Array[(-1)^(# + 1)*psi[#] &, 50]]]
%o A357820 (PARI) f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
%o A357820 a(n) = numerator(sum(k=1, n, (-1)^(k+1)/f(k))); \\ _Michel Marcus_, Oct 15 2022
%Y A357820 Cf. A001615, A173290, A357821 (denominators).
%Y A357820 Cf. A001620, A065463, A335707.
%Y A357820 Similar sequence: A211177.
%K A357820 nonn,frac
%O A357820 1,2
%A A357820 _Amiram Eldar_, Oct 14 2022