

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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


LINKS



FORMULA



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


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



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



