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A033949 Positive integers that do not have a primitive root. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

REFERENCES

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

Brett A. Harrison, On the reducibility of cyclotomic polynomials over finite fields, Amer. Math. Monthly, Vol 114, No. 9 (2007), 813-818

FORMULA

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.

MATHEMATICA

Select[Range[2, 130], !IntegerQ[PrimitiveRoot[#]]&] (* Harvey P. Dale, Oct 25 2011 *)

PROG

(Sage) print [n for n in range(1, 100) if not Integers(n).multiplicative_group_is_cyclic()] # Ralf Stephan, Mar 30 2014

CROSSREFS

Cf. A033948.

Sequence in context: A032455 A050275 * A175594 A062373 A180690 A194592

Adjacent sequences:  A033946 A033947 A033948 * A033950 A033951 A033952

KEYWORD

nonn

AUTHOR

Calculated by Jud McCranie

STATUS

approved

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Last modified September 2 04:45 EDT 2014. Contains 246321 sequences.