OFFSET
1,1
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.
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.
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Sep 29 2014
STATUS
approved