A349530 -id:A349530 - cp's OEIS frontend

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

1, 4, 7, 7, 13, 13, 13, 13, 13, 31, 41, 41, 61, 61, 61, 61, 61, 61, 61, 73, 101, 137, 137, 137, 137, 137, 137, 137, 137, 233, 233, 233, 233, 233, 233, 349, 349, 349, 349, 349, 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

Offset: 1

Zhi-Wei Sun, Nov 21 2021

nonn

Conjecture: a(n) is prime for each n > 2.

We have verified this for n up to 10^4.

			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.

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.

Cf. A000040, A000290, A001093, A208643, A349530, A349459.

f[k_]:=f[k]=33*k^2*(k^3+1);
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,1,70}];Print[tab]

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

Original entry on oeis.org

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

Zhi-Wei Sun, Sep 07 2022

nonn

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.

			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.

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.

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

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]

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

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

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cp's OEIS Frontend

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

Views

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Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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