A349537 Least positive integer m such that the n numbers 33*k^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

1,2

Comments

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

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

Links

Table of n, a(n) for n=1..70.

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.

Example

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.

Mathematica

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]

Crossrefs

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

Sequence in context: A194121 A063194 A071529 * A271919 A201125 A154922

Adjacent sequences: A349534 A349535 A349536 * A349538 A349539 A349540

Keyword

nonn

Author

Zhi-Wei Sun, Nov 21 2021

Status

approved