A392775 - OEIS

%I #12 Jan 25 2026 19:23:58

%S 1,3,5,5,17,17,17,17,17,17,17,17,17,17,17,17,83,83,83,83,83,107,107,

%T 107,107,107,107,167,167,167,167,167,167,167,167,167,167,167,167,167,

%U 167,167,179,179,179,227,227,227,227,227,227,227,227,227,227,227,227,257,257,257

%N Least positive integer m such that 2*3^(k*(k-1)/2) (k = 1..n) are pairwise incongruent modulo m.

%C Conjecture: Let c be any integer congruent to 3 or -3 modulo 8. For any positive integer n, let m(c,n) be the least positive integer m such that the n numbers 2*c^(k(k-1)/2) (k = 1..n) are pairwise incongruent modulo m.

%C (i) m(c,n) is prime for any integer n >= |c|. In particular, a(n) is prime for any integer n > 1.

%C (ii) lim m(c,n)/n = 4 as n tends to the infinity.

%C This is a supplement to the conjecture in A392732.

%H Zhi-Wei Sun, <a href="/A392775/b392775.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory, 133 (2013), no. 8, 2794-2812.

%e a(3) = 5 since 2*3^(1*0/2) = 2, 2*3^(2*1/2) = 6 and 2*3^(3*2/2) = 54 are pairwise incongruent modulo 5.

%t T[k_]:=T[k]=k(k-1)/2;

%t tab={};m=1;Do[Label[bb]; If[Length[Union[Table[Mod[2*PowerMod[3,T[k],m],m], {k, 1, n}]]]==n,tab=Append[tab,m]; Goto[aa]]; m=m+1; Goto[bb]; Label[aa], {n,1,60}];Print[tab]

%Y Cf. A000040, A000244, A000217, A208643, A392732.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Jan 22 2026