

A074243


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


4



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
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OFFSET

1,2


COMMENTS

A positive integer n is in the sequence if x^3 (modulo n) describes a bijection from the set [0...n1] 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


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000(a(1) to a(1501) from Zak Seidov)
Dainis Zeps, On Grinbergs' differential geometry and finite fields, University of Latvia (2019).


FORMULA

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


EXAMPLE

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.


MAPLE

N:= 1000: # to get all terms <= N
Primes:= {2, 3} union select(isprime, {seq(6*i+5, i=0..floor((N5)/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 *)


PROG

(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


CROSSREFS

Cf. A045309, A257301.
Sequence in context: A324968 A255059 A057760 * A302539 A322912 A072720
Adjacent sequences: A074240 A074241 A074242 * A074244 A074245 A074246


KEYWORD

easy,nonn


AUTHOR

Jack Brennen, Sep 19 2002


EXTENSIONS

New name from Charles R Greathouse IV, Apr 20 2015


STATUS

approved



