|
| |
|
|
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.
|
|
2
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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.
|
|
|
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 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,frac,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
