A392732 Least positive integer m such that 2^(k*(k-1)/2) (k = 1..n) are pairwise incongruent modulo m.

1, 2, 4, 5, 13, 17, 17, 17, 47, 47, 47, 47, 59, 59, 59, 59, 83, 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, 179, 179, 179, 179, 227, 227, 227, 227, 227, 227, 227, 227, 227, 227, 227, 227, 263, 263

Offset

1,2

Comments

Conjecture: Let c be any integer with |c| > 1, and let D(c,n) be the least positive integer m such that the n numbers c^(k*(k-1)/2) (k = 1..n) are pairwise incongruent modulo m.

(i) We have lim D(c,n)/n = 4 as n tends to the infinity.

(ii) If neither c nor -c is congruent to 3 modulo 8, then D(c,n) is prime for every integer n > |c| + 1.

(iii) If c is congruent to 3 or -3 modulo 8, then the set {D(c,n): n > 0} contains infinitely many powers of 2.

(iv) Let E(c) = {n > 0: D(c,n) is not prime}. Then E(2) = E(-2) = E(4) = E(6) = E(-6) = {1,3}, E(-4) = {1,5}, E(7) = {1,2}, E(-7) = E(8) = E(-8) = E(9) = E(-9) = {1}, E(10) = {1,3,4,5} and E(-10) = {1,3,4}.

See also A392775 for a supplement.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..5790 from Zhi-Wei Sun)

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

Example

a(4) = 5 since 2^(1*0/2) = 1, 2^(2*1/2) = 2, 2^(3*2/2) = 8 and 2^(4*3/2) = 64 are incongruent modulo 5.

Mathematica

f[k_]:=f[k]=2^(k(k-1)/2);
tab={}; m=1; Do[Label[bb]; If[Length[Union[Table[Mod[f[k], m], {k, 1, n}]]]==n, tab=Append[tab, m]; Goto[aa]]; m=m+1; Goto[bb]; Label[aa], {n, 1, 60}]; Print[tab]

Prog

(Python)
from itertools import count
def A392732(n): return next(m for m in count(n) if len({pow(2, k*(k-1)>>1, m) for k in range(1, n+1)})==n) # Chai Wah Wu, Jan 21 2026

Crossrefs

Cf. A000040, A000079, A000217, A208643, A392775.

Sequence in context: A102932 A128457 A139485 * A079407 A078652 A289491

Adjacent sequences: A392729 A392730 A392731 * A392733 A392734 A392735

Keyword

nonn

Author

Zhi-Wei Sun, Jan 21 2026

Status

approved