cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A323669 Decimal expansion of 15/(2*Pi^2) = 1/((4/5)*zeta(2)).

Original entry on oeis.org

7, 5, 9, 9, 0, 8, 8, 7, 7, 3, 1, 7, 5, 3, 3, 2, 8, 5, 8, 2, 9, 0, 9, 5, 9, 7, 4, 0, 7, 2, 9, 5, 7, 2, 9, 1, 7, 8, 2, 6, 9, 0, 8, 1, 0, 0, 4, 1, 8, 4, 9, 1, 1, 6, 3, 4, 2, 0, 6, 7, 7, 3, 9, 2, 0, 6, 2, 9, 8, 4, 0, 7, 2, 1, 6, 7, 6, 5
Offset: 0

Views

Author

Wolfdieter Lang, Sep 03 2019

Keywords

Comments

This is the limit n -> infinity of (1/n^2)*Phi_1(n) = (1/n^2)*Sum_{k=1..n} psi(k), with Dedekind's psi function psi(k) = k*Product_{p|k} (1 + 1/p) = A001615(k). Distinct primes p dividing k appear, and the empty product for k = 1 is set to 1. See the Walfisz reference, Satz 2., p. 100 (with x -> n, and phi_1(n) = psi(n)).
For the rationals r(n) = (1/n^2)*Phi_1(n) see A327340(n)/A327341(n), n >= 1.

Examples

			0.7599088773175332858290959740729572917826908100418491163420677392062984...

References

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

Links

Crossrefs

Cf. A001615, A059956 (1/zeta(2)), A327340, A327341.

Programs

Formula

Equal to 15/(2*Pi^2) = 1/((4/5)*zeta(2)), with 1/zeta(2) = A059956.

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.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Sep 03 2019

Keywords

Comments

The corresponding numerators are given in A327340.
For details see A327340, also for the Dedekind's psi function, the rationals and the limit.

Examples

			See A327340.

References

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

Crossrefs

Cf. A327340.

Programs

  • 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 *)
  • 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

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.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.