A327340 Numerator 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, 8, 7, 4, 8, 40, 13, 64, 41, 94, 59, 132, 39, 4, 51, 222, 43, 278, 157, 346, 191, 406, 227, 484, 263, 562, 305, 640, 178, 24, 99, 280, 447, 942, 169, 1052, 278, 1168, 31, 1282, 689, 1422, 747, 58, 819, 1686, 99, 1838, 482

Offset

1,3

Comments

The corresponding denominators are given in A327341.

Dedekind's psi(k) = k*Product_{p|k}(1 + 1/p), with primes p, and the empty product is set to 1. See psi(k) = A001615(k), k >= 1. In the Walfisz reference psi(k) = phi_1(k).

In the Walfisz reference, Satz 2., p. 100, the approximation for Phi_1(x) = (15/(2*Pi^2))*x^2 + O(x*(log(x))^{2/3}) is given (with B instead of the O() notation). For the constant 15/(2*Pi^2) see A323669 .

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..50.

Eric Weisstein's World of Mathematics, Dedekind Function.

Wikipedia, Dedekind psi function.

Formula

a(n) = numerator(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 empty product set to 1 for k = 1), for n >= 1.

a(n) = numerator(A173290(n)/n^2). - Amiram Eldar, Nov 24 2022

Example

The rationals (in lowest terms) begin: 1/1, 1/1, 8/9, 7/8, 4/5, 8/9, 40/49, 13/16, 64/81, 41/50, 94/121, 59/72, 132/169, 39/49, 4/5, 51/64, 222/289, 43/54, 278/361, 157/200, 346/441, 191/242, 406/529, 227/288, 484/625, 263/338, 562/729, 305/392, 640/841, 178/225, 24/31, ...
The limit of r(n) for n-> infinity is A323669 = 0.759908877317533285829...
r(10^5) is approximatly 0.7599142240 (10 digits).

Mathematica

psi[0] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); a[n_] := Numerator[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) = numerator(sum(k=1, n, dpsi(k))/n^2); \\ Michel Marcus, Sep 18 2023

Crossrefs

Cf. A001615, A173290, A323669, A327341 (denominators).

Sequence in context: A070702 A019870 A019903 * A167222 A076417 A114137

Adjacent sequences: A327337 A327338 A327339 * A327341 A327342 A327343

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Sep 03 2019

Status

approved