login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A033949 Positive integers that do not have a primitive root. 34
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 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

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 *)

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: A279963 A050275 * A175594 A272592 A062373 A180690

Adjacent sequences:  A033946 A033947 A033948 * A033950 A033951 A033952

KEYWORD

nonn,changed

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 28 04:27 EDT 2017. Contains 287212 sequences.