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

Author

Reinhard Zumkeller, Aug 21 2005

Keywords

• nonn
• frac

Comments

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

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

Examples

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

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)

Crossrefs

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

Programs

Magma

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

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 *)