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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. 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 (list; graph; refs; listen; history; text; internal format)
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 1-4 on Nov. 16, 2021, and verified them for n up to 10^5.

LINKS

Table of n, a(n) for n=1..80.

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) = 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

Zhi-Wei Sun, Sep 07 2022

STATUS

approved

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Last modified November 27 22:56 EST 2022. Contains 358406 sequences. (Running on oeis4.)