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

1, 9, 1393, 257875, 47463376609, 940908897061, 972213062238348973121, 7727182467755471289426059, 10338014371627802833957102351534201, 26038773205374138944970092886340352227, 205885410277133543091182509665217407908365393153956577

Offset

2,2

Comments

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.

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, ...}.

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..49

Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Eric Weisstein's World of Mathematics, Harmonic Number

Formula

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

Mathematica

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

Crossrefs

Cf. A045323, A120289, A120352, A119722.

Sequence in context: A117053 A270067 A213448 * A230172 A231191 A217267

Adjacent sequences: A120344 A120345 A120346 * A120348 A120349 A120350

Keyword

nonn,frac

Author

Alexander Adamchuk, Aug 16 2006, Oct 31 2006

Status

approved