A392775 Least positive integer m such that 2*3^(k*(k-1)/2) (k = 1..n) are pairwise incongruent modulo m.

1, 3, 5, 5, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 83, 83, 83, 83, 83, 107, 107, 107, 107, 107, 107, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 179, 179, 179, 227, 227, 227, 227, 227, 227, 227, 227, 227, 227, 227, 227, 257, 257, 257

Offset

1,2

Comments

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.

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

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

This is a supplement to the conjecture in A392732.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, On functions taking only prime values, J. Number Theory, 133 (2013), no. 8, 2794-2812.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A137780 A079372 A055382 * A165235 A370553 A200771

Adjacent sequences: A392772 A392773 A392774 * A392776 A392777 A392778

Keyword

nonn

Author

Zhi-Wei Sun, Jan 22 2026

Status

approved