This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A033949 Positive integers that do not have a primitive root. 35
 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. See A281854 for the groups. - Wolfdieter Lang, Feb 04 2017 Numbers n with the property that there exists a natural number m with 1A002322(n), or equivalently A034380(n)>1. - Ivan Neretin, Mar 28 2015 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 Wikipedia, Primitive root modulo n 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. MAPLE 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 MATHEMATICA Select[Range[2, 130], !IntegerQ[PrimitiveRoot[#]]&] (* Harvey P. Dale, Oct 25 2011 *) a[n_] := Module[{j, l = {}}, While[Length[l] CarmichaelLambda[j], AppendTo[l, j]; Break[]]]]; l[[n]]]; Array[a, 100] (* Jean-François Alcover, May 29 2018, after Alois P. Heinz's Maple code for A277915 *) PROG (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 CROSSREFS Cf. A033948, A193305 (composites with primitive root). Column k=1 of A277915, A281854. Sequence in context: A032455 A279963 A050275 * A175594 A272592 A062373 Adjacent sequences:  A033946 A033947 A033948 * A033950 A033951 A033952 KEYWORD nonn AUTHOR Calculated by Jud McCranie STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 15 09:22 EDT 2019. Contains 328026 sequences. (Running on oeis4.)