A327341 Denominators of the rationals r(n) = (1/n^2)*Phi_1(n), with Phi_1(n) = Sum{k=1..n} psi(k), with Dedekind's psi function.

1, 1, 9, 8, 5, 9, 49, 16, 81, 50, 121, 72, 169, 49, 5, 64, 289, 54, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 225, 31, 128, 363, 578, 1225, 216, 1369, 361, 1521, 40, 1681, 882, 1849, 968, 75, 1058, 2209, 128, 2401

Offset

1,3

Comments

The corresponding numerators are given in A327340.

For details see A327340, also for the Dedekind's psi function, the rationals and the limit.

References

Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 100, Satz 2.

Links

Table of n, a(n) for n=1..49.

Formula

a(n) = denominator(r(n)), with the rationals r(n) = (1/n^2)*Sum{k=1..n}(k*Product_{p|k}(1 + 1/p)), with distinct prime p divisors of k (with the empty product set to 1 for k = 1), for n >= 1.

Example

See A327340.

Mathematica

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); a[n_] := Denominator[Sum[psi[k], {k, 1, n}]/n^2]; Array[a, 50] (* Amiram Eldar, Sep 03 2019 *)

Prog

(PARI) dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = denominator(sum(k=1, n, dpsi(k))/n^2); \\ Michel Marcus, Sep 18 2023

Crossrefs

Cf. A327340.

Sequence in context: A377602 A200292 A155791 * A059068 A059069 A084660

Adjacent sequences: A327338 A327339 A327340 * A327342 A327343 A327344

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Sep 03 2019

Status

approved