A327341 - OEIS

%I #18 Sep 18 2023 09:01:05

%S 1,1,9,8,5,9,49,16,81,50,121,72,169,49,5,64,289,54,361,200,441,242,

%T 529,288,625,338,729,392,841,225,31,128,363,578,1225,216,1369,361,

%U 1521,40,1681,882,1849,968,75,1058,2209,128,2401

%N 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.

%C The corresponding numerators are given in A327340.

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

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

%F 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.

%e See A327340.

%t 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 *)

%o (PARI) dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615

%o a(n) = denominator(sum(k=1, n, dpsi(k))/n^2); \\ _Michel Marcus_, Sep 18 2023

%Y Cf. A327340.

%K nonn,frac,easy

%O 1,3

%A _Wolfdieter Lang_, Sep 03 2019