A120347 Numerator of Sum_{k=1..n-1} 1/k^n.

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

Offset: 2

Author

Alexander Adamchuk, Aug 16 2006, Oct 31 2006

Keywords

• nonn
• frac

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_{i=1..k} 1/i^r is the generalized harmonic number.

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

Crossrefs

Cf. A045323, A120289, A120352 (a(prime(n))), A119722 (a(prime(n))/prime(n)^3).

Programs

Mathematica

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

Formula

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