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

3,3

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.

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A338102 A167445 A302531 * A303412 A209548 A302898

Adjacent sequences: A060999 A061000 A061001 * A061003 A061004 A061005

Keyword

nonn,easy

Author

N. J. A. Sloane, May 15 2001

Status

approved