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

Author

Amiram Eldar, Oct 14 2022

Keywords

• nonn
• frac

Examples

			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, ...
		

Links

• 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.

Crossrefs

Cf. A001615, A173290, A357819 (denominators).

Cf. A001620, A065463, A335707.

Similar sequences: A028415, A104528, A212717.

Programs

Mathematica

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

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

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).