A061002 As p runs through the primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k } / p^2.

1, 1, 61, 509, 8431, 39541, 36093, 375035183, 9682292227, 40030624861, 1236275063173, 6657281227331, 2690511212793403, 5006621632408586951, 73077117446662772669, 4062642402613316532391, 46571842059597941563297, 8437878094593961096374353

Offset: 3

Author

N. J. A. Sloane, May 15 2001

Keywords

• nonn
• easy

Comments

This is an integer by a theorem of Waring and Wolstenholme.

Conjecture: If p is the n-th prime and H(n) is the n-th harmonic number, then denominator(H(p)/H(p-1))/numerator(H(p-1)/p^2) = p^3. A193758(p)/a(n) = p^3, p > 3. - Gary Detlefs, Feb 20 2013

The sequence which gives the numerators of H_{p-1} = Sum_{k=1..p-1} 1/k for p prime >= 5 is A076637. - Bernard Schott, Dec 02 2018

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 388 Problem 5.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 115.

Links

• Muniru A Asiru, Table of n, a(n) for n = 3..340
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Crossrefs

Cf. A000040, A001008, A001248, A120285, A185399, A193758.

Programs

GAP

List(List(Filtered([5..80],p->IsPrime(p)),i->Sum([1..i-1],k->1/k)/i^2),NumeratorRat); # Muniru A Asiru, Dec 02 2018

Maple

A061002:=proc(n) local p;
  p:=ithprime(n);
  (1/p^2)*numer(add(1/i,i=1..p-1));
end proc;
[seq(A061002(n),n=3..20)];

Mathematica

Table[Function[p, Numerator[Sum[1/k, {k, p - 1}]/p^2]]@ Prime@ n, {n, 3, 20}] (* Michael De Vlieger, Feb 04 2017 *)

PARI

a(n) = my(p=prime(n)); numerator(sum(k=1, p-1, 1/k))/p^2; \\ Michel Marcus, Dec 03 2018

Formula

a(n) = A001008(p-1)/p^2, p=A000040(n). - R. J. Mathar, Jan 09 2017

a(n) = A120285(n)/A001248(n). - R. J. Mathar, Jan 09 2017