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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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62
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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
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OFFSET
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1,2
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COMMENTS
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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
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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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