A033948 Numbers that have a primitive root (the multiplicative group modulo n is cyclic).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 50, 53, 54, 58, 59, 61, 62, 67, 71, 73, 74, 79, 81, 82, 83, 86, 89, 94, 97, 98, 101, 103, 106, 107, 109, 113, 118, 121, 122, 125, 127, 131, 134, 137, 139

Offset

1,2

Comments

The sequence consists of 1, 2, 4 and numbers of the form p^i and 2p^i, where p is an odd prime and i >= 1.

Sequence gives values of n such that x^2 == 1 (mod n) has no solution with 1 < x < n-1. - Benoit Cloitre, Jan 04 2002

Gaussian criterion for terms of the sequence: n is in the sequence iff Product_{1<=i<=n-1, gcd(i,n)=1} i == -1 (mod n), see example. - Vladimir Shevelev, Jan 11 2011

For the criterion used above see the Hardy and Wright reference, Theorem 129. p. 102, a consequence of Bauer's theorem. See also T. D. Noe's comment with the Nagell reference on A060594 and also A160377. - Wolfdieter Lang, Feb 16 2012

Also numbers n such that phi(n) = lambda(n) (or numbers with A034380(n)=1), where phi is A000010, and lambda is Carmichael's lambda: A002322. - Enrique Pérez Herrero, Jun 04 2013

All values of n>2 are given when there are exactly two solutions for n*j+1 is a square, 0 <= j < n, which are j = {0, n-2}. See Mathematica examples. - Richard R. Forberg, Mar 26 2016

Numbers n such that the Galois group of the cyclotomic field with the n-th roots of unity is a cyclic group. [Van der Waerden, p. 55, Th. 4.11.; Corwin, 1967] - N. J. A. Sloane, Nov 26 2016

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 129, p. 102.

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, 4th edition, page 62, Theorem 2.25.

B. L. van der Waerden, Modern Algebra, 2nd. ed., Ungar, NY, Vol. I, 1948.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Anonymous, Notes on Number Theory:Primitive Roots [broken link]

Joerg Arndt, Matters Computational (The Fxtbook), p. 778.

L. J. Corwin, Irreducible polynomials over the integers which factor mod p for every p, Unpublished Bell Labs Memo, Sep 07 1967 [Annotated scanned copy]

Math Reference Project, Primitive Root

Eric Weisstein's World of Mathematics, Primitive Root

Eric Weisstein's World of Mathematics, Modulo Multiplication Group

Wolfram Research, Prime Roots

Example

Gaussian product for n=9 is 1*2*4*5*7*8=2240. Since 2240==-1(mod 9), then 9 is in the sequence. - Vladimir Shevelev, Jan 11 2011

Maple

m := proc(n) local k, r; r := 1; if n = 2 then return false fi;
for k from 1 to n do if igcd(n, k) = 1 then r := modp(r*k, n) fi od; r end:
select(n -> m(n) <> 1, [$1..139]); # Peter Luschny, May 25 2017

Mathematica

Join[{1}, Select[ Range[140], IntegerQ[ PrimitiveRoot[#]] &]] (* Jean-François Alcover, Sep 27 2011 *)
Select[Range[139], EulerPhi[#] == CarmichaelLambda[#] &] (* T. D. Noe, Jun 04 2013 *)
result = {}; Do[count = 0;
Do[If[Mod[j^2, n] == 1, count++], {j, 2, n - 2}];
If[count == 0, AppendTo[result, n]], {n, 1, 200}]; result (* Richard R. Forberg, Mar 26 2016 *)
result = {}; Do[count = 0;
Do[ r = Sqrt[n*j + 1]; If[IntegerQ[r], count++], {j, 0, n}];
If[count == 2, AppendTo[result, n]], {n, 0, 200}]; result  (* missing{1, 2} Richard R. Forberg, Mar 26 2016 *)

Prog

(PARI) is(n)=if(n%2, isprimepower(n) || n==1, n==2 || n==4 || (isprimepower(n/2, &n) && n>2)) \\ Charles R Greathouse IV, Apr 16 2015

Crossrefs

Cf. A033949 (complement), A072209, A001783 (Gaussian products used in the V. Shevelev example).

Cf. also A002322, A060594, A062373, A034380, A160377.

Union of 1, 2, 4, A061345, A278568.

Sequence in context: A048627 A152757 A062462 * A285514 A117730 A193838

Adjacent sequences: A033945 A033946 A033947 * A033949 A033950 A033951

Keyword

nonn

Author

Calculated by Jud McCranie, entered by N. J. A. Sloane

Status

approved