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

Author

Wolfdieter Lang, Sep 03 2019

Keywords

• nonn
• frac
• easy

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 .

Examples

			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).
		

References

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

Links

• Eric Weisstein's World of Mathematics, Dedekind Function.
• Wikipedia, Dedekind psi function.

Crossrefs

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

Programs

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 *)

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

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