A248004 Least positive integer m with prime(m*n) == 1 (mod m+n).

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, 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, 303, 80, 72, 8, 61, 36

Offset: 1

Author

Zhi-Wei Sun, Sep 29 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) exists for any n > 0. Moreover, a(n) does not exceed n*(n-1)/2 if n > 2.

(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.

Examples

			a(2) = 4 since prime(2*4) = 19 is congruent to 1 modulo 2 + 4 = 6.
a(5146) = 593626 since prime(5146*593626) = prime(3054799396) = 73226821741 is congruent to 1 modulo 5146 + 593626 = 598772.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Crossrefs

Cf. A000040, A247824, A247975.

Programs

Mathematica

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}]
lpim[n_]:=Module[{m=1},While[Mod[Prime[m*n],m+n]!=1,m++];m]; Array[lpim,60] (* Harvey P. Dale, Oct 01 2017 *)