cp's OEIS Frontend

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

%I A061002 #46 Aug 17 2024 16:52:15
%S A061002 1,1,61,509,8431,39541,36093,375035183,9682292227,40030624861,
%T A061002 1236275063173,6657281227331,2690511212793403,5006621632408586951,
%U A061002 73077117446662772669,4062642402613316532391,46571842059597941563297,8437878094593961096374353
%N A061002 As p runs through the primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k } / p^2.
%C A061002 This is an integer by a theorem of Waring and Wolstenholme.
%C A061002 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
%C A061002 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
%D A061002 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 388 Problem 5.
%D A061002 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 115.
%H A061002 Muniru A Asiru, <a href="/A061002/b061002.txt">Table of n, a(n) for n = 3..340</a>
%H A061002 R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%F A061002 a(n) = A001008(p-1)/p^2, p=A000040(n). - _R. J. Mathar_, Jan 09 2017
%F A061002 a(n) = A120285(n)/A001248(n). - _R. J. Mathar_, Jan 09 2017
%p A061002 A061002:=proc(n) local p;
%p A061002   p:=ithprime(n);
%p A061002   (1/p^2)*numer(add(1/i,i=1..p-1));
%p A061002 end proc;
%p A061002 [seq(A061002(n),n=3..20)];
%t A061002 Table[Function[p, Numerator[Sum[1/k, {k, p - 1}]/p^2]]@ Prime@ n, {n, 3, 20}] (* _Michael De Vlieger_, Feb 04 2017 *)
%o A061002 (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
%o A061002 (PARI) a(n) = my(p=prime(n)); numerator(sum(k=1, p-1, 1/k))/p^2; \\ _Michel Marcus_, Dec 03 2018
%Y A061002 Cf. A000040, A001008, A001248, A120285, A185399, A193758.
%K A061002 nonn,easy
%O A061002 3,3
%A A061002 _N. J. A. Sloane_, May 15 2001