A119722 Numerator of generalized harmonic number H(p-1,p)= Sum[ 1/k^p, {k,1,p-1}] divided by p^3 for prime p>3.

2063, 2743174627, 19563315706517008974432827112201617, 2597378078067393746941400113704449589199274012223316613, 777478358612529699991463948563778410644748121498526065585976638854277886379480749840301120148933

Offset: 3

Author

Alexander Adamchuk, Jun 13 2006

Keywords

• frac
• nonn

Comments

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,p) is divisible by p^3 for prime p>3.

Examples

			Prime[3] = 5.
a(3) = numerator[ 1 + 1/2^5 + 1/3^5 + 1/4^5 ] / 5^3 = 257875/125 = 2063.
Prime[4] = 7
a(4) = numerator[ 1 + 1/2^7 + 1/3^7 + 1/4^7 + 1/5^7 + 1/6^7 ] / 7^3 = 2743174627.
		

Links

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

Crossrefs

Cf. A099828, A099827, A001008, A007406, A007408, A007410.

Programs

Mathematica

Numerator[Table[Sum[1/k^Prime[n],{k,1,Prime[n]-1}],{n,3,9}]]/Table[Prime[n]^3,{n,3,9}]

Formula

a(n) = numerator[ Sum[ 1/k^Prime[n], {k,1,Prime[n]-1} ]] / Prime[n]^3 for n>2.