A128820 Numerator of alternating generalized harmonic number H'(p-1,2p) = Sum_{k=1..p-1} (-1)^(k+1)/k^(2*p) divided by p^2 for prime p > 2.

7, 2474315503, 53305712401979540402437, 5597916593064896381208777124641713285719656398067086247546781015747740847, 192635872080422175485338764164035657976855166649911323825254242037669356649787653784405726270977624462974729613783

Offset: 2

Author

Alexander Adamchuk, Apr 10 2007

Keywords

• nonn

Comments

Alternating generalized harmonic number is H'(n,m) = Sum_{k=1..n} (-1)^(k+1)*1/k^m. Numerator of H'(p-1,2n) is divisible by p for all integers n > 0 and primes p > 2. Numerator of H'(p-1,2p) is divisible by p^2 for prime p > 2.

Examples

			prime(2) = 3; a(2) = numerator(1 - 1/2^6) / 3^2 = 63/9 = 7.
prime(3) = 5; a(3) = numerator(1 - 1/2^10 + 1/3^10 - 1/4^10) / 5^2 = 61857887575/25 = 2474315503.
		

Links

• Eric Weisstein's World of Mathematics, Harmonic Number
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Crossrefs

Cf. A119722 (numerator of generalized harmonic number H(p-1, p) = Sum_{k=1..p-1} 1/k^p divided by p^3 for prime p>3).

Cf. A001008, A119682, A120296.

Programs

Mathematica

Table[ Numerator[ Sum[(-1)^(k+1)*1/k^(2*Prime[n]), {k,1,Prime[n]-1} ] ] / Prime[n]^2, {n,2,10} ]

Formula

a(n) = numerator(Sum_{k=1..prime(n)-1} (-1)^(k+1)/k^(2*prime(n))) / prime(n)^2 for n > 1.