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A033948 Numbers that have a primitive root (the multiplicative group modulo n is cyclic). 21
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 (list; graph; refs; listen; history; text; internal format)
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 Prod{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

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.

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

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

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 *)

CROSSREFS

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

Cf. A062373, A034380.

Sequence in context: A048627 A152757 A062462 * A117730 A174328 A123101

Adjacent sequences:  A033945 A033946 A033947 * A033949 A033950 A033951

KEYWORD

nonn

AUTHOR

Calculated by Jud McCranie

STATUS

approved

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Last modified November 23 18:50 EST 2014. Contains 249865 sequences.