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A103349
Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).
4
1, 257, 1686433, 431733409, 168646292872321, 168646392872321, 972213062238348973121, 248886558707571775009601, 1632944749460578249437992161, 1632944765723715465050248417
OFFSET
1,2
COMMENTS
a(n) gives the partial sums, Zeta(8,n) of Euler's Zeta(8). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) A001008/A002805.
For the denominators see A103350 and for the rationals Zeta(8,n) see the W. Lang link under A103345.
FORMULA
a(n)=numerator(sum_{k=1..n} 1/k^8).
G.f. for rationals Zeta(8, n): polylogarithm(8, x)/(1-x).
MATHEMATICA
s=0; lst={}; Do[s+=n^1/n^9; AppendTo[lst, Numerator[s]], {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)
Accumulate[1/Range[10]^8]//Numerator (* Harvey P. Dale, Aug 11 2024 *)
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Feb 15 2005
STATUS
approved