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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...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

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000(a(1) to a(1501) from Zak Seidov)

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((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 *)

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: A076474 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

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)