cp's OEIS Frontend

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

%I A356976 #17 Sep 07 2022 10:23:45
%S A356976 1,3,3,7,15,15,19,27,27,39,39,39,61,61,81,81,81,81,81,81,81,81,81,81,
%T A356976 81,81,81,243,243,243,243,243,243,243,243,243,243,243,243,243,243,243,
%U A356976 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
%N A356976 Least positive integer m such that the numbers k^3 + 3*k (k = 1..n) are pairwise distinct modulo m.
%C A356976 Conjecture 1: If n is at least 15, then a(n) is the least power of 3 not smaller than 3*n.
%C A356976 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.
%C A356976 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.
%C A356976 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.
%C A356976 The author formulated Conjectures 1-4 on Nov. 16, 2021, and verified them for n up to 10^5.
%H A356976 Zhi-Wei Sun, <a href="http://dx.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.
%H A356976 Zhi-Wei Sun, <a href="http://hitpress.hit.edu.cn/2021/1015/c12593a261001/page.htm">New Conjectures in Number Theory and Combinatorics</a> (in Chinese), Harbin Institute of Technology Press, 2021.
%H A356976 Quan-Hui Yang and Lilu Zhao, <a href="http://arxiv.org/abs/2111.02746">On a conjecture of Sun involving powers of three</a>, arXiv:2111.02746 [math.NT], 2021.
%e A356976 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.
%t A356976 f[k_]:=f[k]=k^3+3*k;
%t A356976 U[m_, n_]:=U[m, n]=Length[Union[Table[Mod[f[k], m], {k, 1, n}]]]
%t A356976 tab={}; s=1; Do[m=s; Label[bb]; If[U[m, n]==n, s=m; tab=Append[tab, s]; Goto[aa]];
%t A356976 m=m+1; Goto[bb]; Label[aa], {n, 1, 80}]; Print[tab]
%Y A356976 Cf. A000578, A079908, A349459, A349530, A349537.
%K A356976 nonn
%O A356976 1,2
%A A356976 _Zhi-Wei Sun_, Sep 07 2022