|
%I #14 Oct 01 2017 12:09:53
%S 3,4,1,2,2,15,1,1,5,10,2,3,4,18,6,27,4,7,35,4,45,2,47,9,5,10,16,11,3,
%T 3,9,61,1,52,3,60,53,74,8,47,7,60,128,5,21,12,2,29,15,127,53,28,17,21,
%U 303,80,72,8,61,36
%N Least positive integer m with prime(m*n) == 1 (mod m+n).
%C Conjecture: (i) a(n) exists for any n > 0. Moreover, a(n) does not exceed n*(n-1)/2 if n > 2.
%C (ii) For each positive integer n, there is an integer m > 0 with prime(m*n) == -1 (mod m+n). Moreover, we may require that m does not exceed n*(n-1)/2 if n > 2.
%H Zhi-Wei Sun, <a href="/A248004/b248004.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1409.5685">A new theorem on the prime-counting function</a>, arXiv:1409.5685, 2014.
%e a(2) = 4 since prime(2*4) = 19 is congruent to 1 modulo 2 + 4 = 6.
%e a(5146) = 593626 since prime(5146*593626) = prime(3054799396) = 73226821741 is congruent to 1 modulo 5146 + 593626 = 598772.
%t Do[m=1;Label[aa];If[Mod[Prime[m*n],m+n]==1,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
%t lpim[n_]:=Module[{m=1},While[Mod[Prime[m*n],m+n]!=1,m++];m]; Array[lpim,60] (* _Harvey P. Dale_, Oct 01 2017 *)
%Y Cf. A000040, A247824, A247975.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Sep 29 2014
|
