A111918 Numerator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^3)).

1, 9, 89, 721, 18601, 2089, 103961, 832913, 68093153, 68347169, 8320810649, 8331482849, 1414167788681, 1416817979081, 1421435199689, 11373510649537, 3295255574810593, 366551352989977, 132591913780524097, 132652127531625601

Offset

1,2

Comments

Denominator of x(n) = A111919(n);

x(n) = a(n)/A111919(n) ---> Pi*Pi/7 = 6*zeta(2)/7.

References

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Sect. 6, Problem 50.

Links

Robert Israel, Table of n, a(n) for n = 1..1150

Eric Weisstein's World of Mathematics, Odd Part

Eric Weisstein's World of Mathematics, Riemann Zeta Function zeta(2)

Example

a(50) = 429245027972423430658635002176171233144054521,
A111919(50) = 307330458857514095936081844184308729630720000:
x(50) = a(50)/A111919(50) = 1.39668..., x(50)*7/6 = 1.62946....

Maple

S:= 0: Res:= NULL:
for k from 1 to 25 do
  S:= S + 1/k^2/2^padic:-ordp(k, 2);
  Res:= Res, numer(S)
od:
Res; # Robert Israel, Jan 13 2020

Mathematica

oddPart[n_] := n/2^IntegerExponent[n, 2];
x[n_] := Sum[oddPart[k]/k^3, {k, 1, n}];
a[n_] := Numerator[x[n]];
Array[a, 20] (* Jean-François Alcover, Dec 13 2021 *)

Prog

(Magma) val:=func<n|n/2^Valuation(n, 2)>; [Numerator(&+[val(k)/(k^3):k in [1..n]]):n in [1..20]]; // Marius A. Burtea, Jan 13 2020

Crossrefs

Cf. A000265, A013661, A111919, A111929, A111920, A111922.

Sequence in context: A175371 A291893 A217321 * A064616 A133486 A224760

Adjacent sequences: A111915 A111916 A111917 * A111919 A111920 A111921

Keyword

nonn,frac

Author

Reinhard Zumkeller, Aug 21 2005

Status

approved