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A349537 Least positive integer m such that the n numbers 33*k^2*(k^3+1) (k = 1..n) are pairwise distinct modulo m.

%I A349537 #7 Nov 27 2021 06:46:58
%S A349537 1,4,7,7,13,13,13,13,13,31,41,41,61,61,61,61,61,61,61,73,101,137,137,
%T A349537 137,137,137,137,137,137,233,233,233,233,233,233,349,349,349,349,349,
%U A349537 349,349,349,349,349,349,349,349,349,349,349,349,349,349,547,547,547,547,547,547,547,547,547,547,859,859,859,859,859,859
%N A349537 Least positive integer m such that the n numbers 33*k^2*(k^3+1) (k = 1..n) are pairwise distinct modulo m.
%C A349537 Conjecture: a(n) is prime for each n > 2.
%C A349537 We have verified this for n up to 10^4.
%H A349537 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 A349537 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 A349537 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 A349537 a(2) = 4 since 33*1^2*(1^3+1) = 66 and 33*2^2*(2^3+1) = 1188 are incongruent modulo 4, but they are congruent modulo each of 1, 2 and 3.
%t A349537 f[k_]:=f[k]=33*k^2*(k^3+1);
%t A349537 U[m_,n_]:=U[m,n]=Length[Union[Table[Mod[f[k],m],{k,1,n}]]]
%t A349537 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,1,70}];Print[tab]
%Y A349537 Cf. A000040, A000290, A001093, A208643, A349530, A349459.
%K A349537 nonn
%O A349537 1,2
%A A349537 _Zhi-Wei Sun_, Nov 21 2021