A356976 Least positive integer m such that the numbers k^3 + 3*k (k = 1..n) are pairwise distinct modulo m.

1, 3, 3, 7, 15, 15, 19, 27, 27, 39, 39, 39, 61, 61, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243, 243

Offset: 1

Author

Zhi-Wei Sun, Sep 07 2022

Keywords

• nonn

Comments

Conjecture 1: If n is at least 15, then a(n) is the least power of 3 not smaller than 3*n.

Conjecture 2: For each positive integer n, the least positive integer m such that those numbers 2*k^3 + k (k = 1..n) are pairwise distinct modulo m, is just the least power of 2 not smaller than n.

Conjecture 3: For any positive integer n, the least positive integer m such that those numbers 2*k^3 - 4*k (k = 1..n) are pairwise distinct modulo m, is just the least power of 3 not smaller than n.

Conjecture 4: For each positive integer n not equal to 4, the least positive integer m such that those numbers 16*k^3 - 8*k (k = 1..n) are pairwise distinct modulo m, is just the least power of 3 not smaller than n.

The author formulated Conjectures 1-4 on Nov. 16, 2021, and verified them for n up to 10^5.

Examples

			a(2) = 3, for, 1^3 + 3*1 = 4 and 2^3 + 3*2 = 14 are incongruent modulo 3, but congruent modulo 1 and 2.
		

Links

• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
• Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.
• Quan-Hui Yang and Lilu Zhao, On a conjecture of Sun involving powers of three, arXiv:2111.02746 [math.NT], 2021.

Crossrefs

Cf. A000578, A079908, A349459, A349530, A349537.

Programs

Mathematica

f[k_]:=f[k]=k^3+3*k;
U[m_, n_]:=U[m, n]=Length[Union[Table[Mod[f[k], m], {k, 1, n}]]]
tab={}; s=1; Do[m=s; Label[bb]; If[U[m, n]==n, s=m; tab=Append[tab, s]; Goto[aa]];
m=m+1; Goto[bb]; Label[aa], {n, 1, 80}]; Print[tab]