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A033948
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Numbers that have a primitive root (the multiplicative group modulo n is cyclic).
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16
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| Sequence gives values of n such that x^2 == 1 (mod n) has no solution with 1<x<n-1. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 04 2002
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REFERENCES
| I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, 4th edition, page 62, Theorem 2.25.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
Anonymous, Notes on Number Theory:Primitive Roots [broken link]
Math Reference Project, Primitive Root
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wolfram Research, Prime Roots
Eric Weisstein's World of Mathematics, Modulo Multiplication Group
Joerg Arndt, Fxtbook, p.778
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FORMULA
| 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.
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) - Vladimir Shevelev(shevelev(AT)bgu.ac.il) 11 Jan 2011
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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.
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MATHEMATICA
| Join[{1}, Select[ Range[140], IntegerQ[ PrimitiveRoot[#]] &]] (* From Jean-François Alcover, Sep 27 2011 *)
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CROSSREFS
| Cf. A033949 (complement), A072209.
Sequence in context: A048627 A152757 A062462 * A117730 A174328 A123101
Adjacent sequences: A033945 A033946 A033947 * A033949 A033950 A033951
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KEYWORD
| nonn
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AUTHOR
| Calculated by Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu)
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