A357818 - OEIS

%I #11 Oct 15 2022 07:17:25

%S 1,4,19,7,23,2,17,53,55,169,175,89,641,1303,331,1345,1373,1387,7061,

%T 2377,9613,29119,29539,29749,6017,6065,6121,6163,31151,31291,15803,

%U 3977,16013,48319,24317,12211,233899,58774,472757,59344,119543,1918673,21249043,21336823

%N Numerators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 100, p. 169.

%H V. Sita Ramaiah and D. Suryanarayana, <a href="http://doi.org/10.18926/mjou/33820">Sums of reciprocals of some multiplicative functions</a>, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.

%H 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 a(n) = numerator(Sum_{k=1..n} 1/psi(k)).

%F a(n)/A357819(n) ~ C * (log(n) + gamma + D) + 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) (Sita Ramaiah and Suryanarayana, 1979; Tóth, 2017).

%e Fractions begin with 1, 4/3, 19/12, 7/4, 23/12, 2, 17/8, 53/24, 55/24, 169/72, 175/72, 89/36, ...

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

%o (PARI) f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615

%o a(n) = numerator(sum(k=1, n, 1/f(k))); \\ _Michel Marcus_, Oct 15 2022

%Y Cf. A001615, A173290, A357819 (denominators).

%Y Cf. A001620, A065463, A335707.

%Y Similar sequences: A028415, A104528, A212717.

%K nonn,frac

%O 1,2

%A _Amiram Eldar_, Oct 14 2022