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

%I A074243 #51 Apr 21 2025 08:29:58
%S A074243 1,2,3,5,6,10,11,15,17,22,23,29,30,33,34,41,46,47,51,53,55,58,59,66,
%T A074243 69,71,82,83,85,87,89,94,101,102,106,107,110,113,115,118,123,131,137,
%U A074243 138,141,142,145,149,159,165,166,167,170,173,174,177,178,179,187,191,197
%N A074243 Numbers n such that every integer has a cube root mod n.
%C A074243 A positive integer n is in the sequence if x^3 (modulo n) describes a bijection from the set [0...n-1] to itself.
%C A074243 Every member of the sequence is squarefree. If m and n are coprime members of the sequence, m*n is also a member.
%C A074243 All primes > 3 in this sequence are congruent to 5 mod 6. See A045309. - _Zak Seidov_, Feb 16 2013
%C A074243 Products of distinct members of A045309 (primes not 1 mod 3). - _Charles R Greathouse IV_, Apr 20 2015
%C A074243 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
%H A074243 Robert Israel, <a href="/A074243/b074243.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1501 from Zak Seidov).
%H A074243 Olivier Garet, <a href="https://arxiv.org/abs/2504.13511">How often is x->x^3 one-to-one in Z/nZ?</a>, arXiv:2504.13511 [math.NT], 2025.
%H A074243 Dainis Zeps, <a href="https://doi.org/10.13140/RG.2.2.28885.27367">On Grinbergs' differential geometry and finite fields</a>, University of Latvia (2019).
%F A074243 a(n) ~ k*n*sqrt(log(n)) for some constant k. - _Charles R Greathouse IV_, Apr 20 2015
%e A074243 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.
%p A074243 N:= 1000: # to get all terms <= N
%p A074243 Primes:= {2,3} union select(isprime, {seq(6*i+5,i=0..floor((N-5)/6))}):
%p A074243 A:= {1}:
%p A074243 for p in Primes do
%p A074243 A:= A union map(`*`, select(`<=`, A, floor(N/p)),p)
%p A074243 od:
%p A074243 A;
%p A074243 # if using Maple 11 or earlier, uncomment the next line
%p A074243 # sort(convert(A,list)); # _Robert Israel_, Apr 20 2015
%t A074243 fQ[n_] := Sort[PowerMod[#, 3, n] & /@ Range@ n] == Range@ n - 1; Select[Range@ 200, fQ] (* _Michael De Vlieger_, Apr 20 2015 *)
%o A074243 (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
%Y A074243 Cf. A045309, A257301.
%K A074243 easy,nonn
%O A074243 1,2
%A A074243 _Jack Brennen_, Sep 19 2002
%E A074243 New name from _Charles R Greathouse IV_, Apr 20 2015