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A120347 Numerator of Sum[ 1/k^n, {k,1,n-1} ]. 2

%I

%S 1,9,1393,257875,47463376609,940908897061,972213062238348973121,

%T 7727182467755471289426059,10338014371627802833957102351534201,

%U 26038773205374138944970092886340352227,205885410277133543091182509665217407908365393153956577

%N Numerator of Sum[ 1/k^n, {k,1,n-1} ].

%C Prime p>2 divides a(p). p^3 divides a(p) for prime p>3. p divides a((p+1)/2) for prime p = {7,11,17,19,23,31,41,43,47,59,67,71,73,79,83,89,97,103,...} = all primes excluding 2 and 3 from A045323[n] Primes congruent to {1, 2, 3, 7} mod 8.

%C a(n) = Numerator[ H(n-1,n) ], where H(k,r)= Sum[ 1/i^r, {i,1,k} ] is generalized harmonic number. Numerators of Sum[ 1/k^p, {k,1,p-1} ]], where p = Prime[n], are listed in A120352(n) = {1, 9, 257875, 940908897061, 26038773205374138944970092886340352227, ...}. a(p)/p^3 for prime p>3 are listed in A119722(n) = {2063, 2743174627, 19563315706517008974432827112201617, ...}.

%H Vincenzo Librandi, <a href="/A120347/b120347.txt">Table of n, a(n) for n = 2..49</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>

%F a(n) = Numerator[Sum[1/k^n,{k,1,n-1}]]. a(n) = Numerator[Zeta[n] - Zeta[n,n]].

%t Table[Numerator[Sum[1/k^n,{k,1,n-1}]],{n,2,15}]

%Y Cf. A045323, A120289, A120352, A119722.

%K nonn,frac

%O 2,2

%A _Alexander Adamchuk_, Aug 16 2006, Oct 31 2006

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Last modified June 23 08:25 EDT 2021. Contains 345395 sequences. (Running on oeis4.)