A357818 Numerators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

1, 4, 19, 7, 23, 2, 17, 53, 55, 169, 175, 89, 641, 1303, 331, 1345, 1373, 1387, 7061, 2377, 9613, 29119, 29539, 29749, 6017, 6065, 6121, 6163, 31151, 31291, 15803, 3977, 16013, 48319, 24317, 12211, 233899, 58774, 472757, 59344, 119543, 1918673, 21249043, 21336823

Offset

1,2

Links

Table of n, a(n) for n=1..44.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 100, p. 169.

V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.

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/psi(k)).

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).

Example

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, ...

Mathematica

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

Prog

(PARI) f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = numerator(sum(k=1, n, 1/f(k))); \\ Michel Marcus, Oct 15 2022

Crossrefs

Cf. A001615, A173290, A357819 (denominators).

Cf. A001620, A065463, A335707.

Similar sequences: A028415, A104528, A212717.

Sequence in context: A222323 A012496 A024572 * A009267 A009275 A362249

Adjacent sequences: A357815 A357816 A357817 * A357819 A357820 A357821

Keyword

nonn,frac

Author

Amiram Eldar, Oct 14 2022

Status

approved