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A033949
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Positive integers that do not have a primitive root.
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34
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8, 12, 15, 16, 20, 21, 24, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116, 117, 119, 120, 123
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OFFSET
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1,1
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COMMENTS
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Numbers n such that the cyclotomic polynomial Phi(n,x) is reducible over Zp for all primes p. Harrison shows that this is equivalent to n>2 and the discriminant of Phi(n,x), A004124(n), being a square. - T. D. Noe, Nov 06 2007
The multiplicative group modulo n is non-cyclic. See the complement A033948. - Wolfdieter Lang, Mar 14 2012. See A281854 for the groups. - Wolfdieter Lang, Feb 04 2017
Numbers n with the property that there exists a natural number m with 1<m<n-1 and m^2 == 1 mod n. - Reinhard Muehlfeld, May 27 2014
Also, numbers n for which A000010(n)>A002322(n), or equivalently A034380(n)>1. - Ivan Neretin, Mar 28 2015
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REFERENCES
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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
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T. D. Noe, Table of n, a(n) for n = 1..10000
Brett A. Harrison, On the reducibility of cyclotomic polynomials over finite fields, Amer. Math. Monthly, Vol 114, No. 9 (2007), 813-818
Wikipedia, Primitive root modulo n
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FORMULA
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Positive integers except 1, 2, 4 and numbers of the form p^i and 2p^i, where p is an odd prime and i >= 1.
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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:
select(n -> m(n) = 1, [$1..123]); # Peter Luschny, May 25 2017
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MATHEMATICA
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Select[Range[2, 130], !IntegerQ[PrimitiveRoot[#]]&] (* Harvey P. Dale, Oct 25 2011 *)
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PROG
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(Sage) print [n for n in range(1, 100) if not Integers(n).multiplicative_group_is_cyclic()] # Ralf Stephan, Mar 30 2014
(Haskell)
a033949 n = a033949_list !! (n-1)
a033949_list = filter
(\x -> any ((== 1) . (`mod` x) . (^ 2)) [2 .. x-2]) [1..]
-- Reinhard Zumkeller, Dec 10 2014
(PARI) is(n)=n>7 && (!isprimepower(if(n%2, n, n/2)) || n>>valuation(n, 2)==1) \\ Charles R Greathouse IV, Oct 08 2016
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CROSSREFS
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Cf. A033948. A193305 (composites with primitive root).
Column k=1 of A277915, A281854.
Sequence in context: A279963 A050275 * A175594 A272592 A062373 A180690
Adjacent sequences: A033946 A033947 A033948 * A033950 A033951 A033952
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KEYWORD
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nonn
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AUTHOR
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Calculated by Jud McCranie
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STATUS
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approved
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