cp's OEIS Frontend

A232862 Least positive integer m <= n^2/2 + 3 such that the set {prime(k) - k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

%I A232862 #28 Jan 04 2018 11:55:01
%S A232862 1,3,4,11,9,8,10,15,29,13,23,22,23,37,32,28,48,44,53,41,45,67,76,117,
%T A232862 119,91,121,88,89,101,72,88,100,143,144,185,145,104,176,141,144,175,
%U A232862 187,213,121,255,128,129,189,243,122,267,275,242,209,205,130,153,263,335
%N A232862 Least positive integer m <= n^2/2 + 3 such that the set {prime(k) - k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.
%C A232862 Conjecture: a(n) > 0 for all n > 0.
%C A232862 Note that a(4) = 11 = 4^2/2 + 3.
%H A232862 Chai Wah Wu, <a href="/A232862/b232862.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..1000 from Zhi-Wei Sun)
%H A232862 Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
%e A232862 a(3) = 4 since prime(1) - 1 = prime(2) - 2 = 1, prime(3) - 3 = 2, prime(4) - 4 = 3, and {1,2,3} is a complete system of residues modulo 3.
%t A232862   L[m_,n_]:=Length[Union[Table[Mod[Prime[k]-k,n],{k,1,m}]]]
%t A232862 Do[Do[If[L[m,n]==n,Print[n," ",m];Goto[aa]],{m,1,n^2/2+3}];
%t A232862 Print[n," ",0];Label[aa];Continue,{n,1,60}]
%Y A232862 Cf. A014689, A232616, A232463, A232548, A232861.
%K A232862 nonn
%O A232862 1,2
%A A232862 _Zhi-Wei Sun_, Dec 01 2013