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A349530 Least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2.

%I A349530 #30 Mar 11 2023 06:55:54
%S A349530 1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,15,15,15,15,15,15,
%T A349530 15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,
%U A349530 15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,25,25,25,25,25
%N A349530 Least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2.
%C A349530 Conjecture 1: Suppose that 5^a < sqrt(n) <= 5^(a+1). Then a(n) = 3*5^a if sqrt(n) <= sqrt(3)*5^a, and a(n) = 5^(a+1) otherwise.
%C A349530 Conjecture 2: Let f(n) be the least positive integer m such that the n numbers 18k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2. Then f(n) is the least power of 5 not smaller than sqrt(n), except for 25 < n <= 45 (and in this case f(n) = 19).
%C A349530 Conjecture 3: Let n be a positive integer not among 26, 27, 28, 626, 627, 628, 629, 630, and define D(n) as the least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m. If 5^a < n <= 3*5^a, then D(n) = 3*5^a. If 3*5^a < n <= 5^(a+1), then D(n) = 5^(a+1).
%C A349530 We have verified the above conjectures for n up to 10^5.
%H A349530 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 A349530 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 A349530 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 A349530 a(2) = 3 since the two numbers 1*(1^4+1) = 2 and 2*(2^4+1) = 34 are distinct modulo 3^2, but they are congruent modulo each of 1^2 and 2^2.
%t A349530 f[k_]:=f[k]=k*(k^4+1);
%t A349530 U[m_,n_]:=U[m,n]=Length[Union[Table[Mod[f[k],m^2],{k,1,n}]]]
%t A349530 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]
%Y A349530 Cf. A000351, A002523, A208643, A349459, A349537.
%K A349530 nonn
%O A349530 1,2
%A A349530 _Zhi-Wei Sun_, Nov 21 2021