A074243 Numbers n such that every integer has a cube root mod n.

1, 2, 3, 5, 6, 10, 11, 15, 17, 22, 23, 29, 30, 33, 34, 41, 46, 47, 51, 53, 55, 58, 59, 66, 69, 71, 82, 83, 85, 87, 89, 94, 101, 102, 106, 107, 110, 113, 115, 118, 123, 131, 137, 138, 141, 142, 145, 149, 159, 165, 166, 167, 170, 173, 174, 177, 178, 179, 187, 191, 197

Offset: 1

Author

Jack Brennen, Sep 19 2002

Keywords

• easy
• nonn

Comments

A positive integer n is in the sequence if x^3 (modulo n) describes a bijection from the set [0...n-1] to itself.

Every member of the sequence is squarefree. If m and n are coprime members of the sequence, m*n is also a member.

All primes > 3 in this sequence are congruent to 5 mod 6. See A045309. - Zak Seidov, Feb 16 2013

Products of distinct members of A045309 (primes not 1 mod 3). - Charles R Greathouse IV, Apr 20 2015

This sequence gives all values, ordered increasingly, for which A257301 vanishes, i.e., A257301(a(n))=0 for any n. - Stanislav Sykora, May 26 2015

Examples

			The number 30 is in the sequence because the function x^3 (mod 30) describes a bijection from [0...29] to itself. Thus every integer has a cube root, modulo 30.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (terms 1..1501 from Zak Seidov).
• Olivier Garet, How often is x->x^3 one-to-one in Z/nZ?, arXiv:2504.13511 [math.NT], 2025.
• Dainis Zeps, On Grinbergs' differential geometry and finite fields, University of Latvia (2019).

Crossrefs

Cf. A045309, A257301.

Programs

Maple

N:= 1000: # to get all terms <= N
Primes:= {2,3} union select(isprime, {seq(6*i+5,i=0..floor((N-5)/6))}):
A:= {1}:
for p in Primes do
A:= A union map(`*`, select(`<=`, A, floor(N/p)),p)
od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 20 2015

Mathematica

fQ[n_] := Sort[PowerMod[#, 3, n] & /@ Range@ n] == Range@ n - 1; Select[Range@ 200, fQ] (* Michael De Vlieger, Apr 20 2015 *)

PARI

is(n)=my(f=factor(n)); if(n>1 && vecmax(f[,2])>1, return(0)); for(i=1,#f~, if(f[i,1]%3==1, return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015

Formula

a(n) ~ k*n*sqrt(log(n)) for some constant k. - Charles R Greathouse IV, Apr 20 2015

Extensions

New name from Charles R Greathouse IV, Apr 20 2015