

A356976


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


0



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
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OFFSET

1,2


COMMENTS

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 14 on Nov. 16, 2021, and verified them for n up to 10^5.


LINKS

Table of n, a(n) for n=1..80.
ZhiWei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 27942812.
ZhiWei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.
QuanHui Yang and Lilu Zhao, On a conjecture of Sun involving powers of three, arXiv:2111.02746 [math.NT], 2021.


EXAMPLE

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.


MATHEMATICA

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]


CROSSREFS

Cf. A000578, A079908, A349459, A349530, A349537.
Sequence in context: A218372 A218242 A218288 * A056420 A030069 A004043
Adjacent sequences: A356973 A356974 A356975 * A356977 A356978 A356979


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Sep 07 2022


STATUS

approved



