cp's OEIS Frontend

A242223 Least prime p such that H(n) == 0 (mod p) but H(k) == 0 (mod p) for no 0 < k < n, or 1 if such a prime p does not exist, where H(n) denotes the n-th harmonic number sum_{k=1..n}1/k.

%I A242223 #16 Aug 05 2019 03:03:54
%S A242223 1,3,11,5,137,7,1,761,7129,61,97,13,29,1049,41233,17,37,19,7440427,
%T A242223 11167027,18858053,23,583859,577,109,34395742267,521,375035183,
%U A242223 4990290163,31,2667653736673,2917,269,3583,397,1297,10839223,199,737281,41
%N A242223 Least prime p such that H(n) == 0 (mod p) but H(k) == 0 (mod p) for no 0 < k < n, or 1 if such a prime p does not exist, where H(n) denotes the n-th harmonic number sum_{k=1..n}1/k.
%C A242223 Conjecture: a(n) is prime except for n = 1, 7.
%H A242223 Zhi-Wei Sun, <a href="/A242223/b242223.txt">Table of n, a(n) for n = 1..184</a>
%H A242223 Z.-W. Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
%e A242223 a(4) = 5 since H(4) = 25/12 == 0 (mod 5), but none of H(1) = 1, H(2) = 3/2 and H(3) = 11/6 is congruent to 0 modulo 5.
%t A242223 h[n_]:=Numerator[HarmonicNumber[n]]
%t A242223 f[n_]:=FactorInteger[h[n]]
%t A242223 p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
%t A242223 Do[If[h[n]<2,Goto[cc]];Do[Do[If[Mod[h[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,40}]
%Y A242223 Cf. A000040, A001008, A002805, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195, A242207, A242222.
%K A242223 nonn
%O A242223 1,2
%A A242223 _Zhi-Wei Sun_, May 08 2014