

A180634


Numbers n such that the discriminant of the nth cyclotomic polynomial is a square.


1



1, 2, 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
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OFFSET

1,2


COMMENTS

A number n is in this sequence if the Galois group of the nth cyclotomic polynomial over the rationals contains only even permutations.
Essentially the same as A033949.  R. J. Mathar, Oct 15 2011
Also, numbers n such that the product of the elements in the group Z_n of invertible elements mod n (i.e., the product mod n of x such that 1 <= x < n and x is coprime to n) is 1. An equivalent characterization of the latter (apart from n=2): n such that the number of square roots of 1 mod n is divisible by 4. (See comments at A033949).  Robert Israel, Dec 08 2014
To see this, use Gauss's generalization of Wilson's theorem namely, the product of the units of Z_n is 1 if n is 4 or p^i or 2p^i for odd primes p, i >0, and is equal to 1 otherwise.  W. Edwin Clark, Dec 09 2014


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.


EXAMPLE

n=5: The 5th cyclotomic polynomial is x^4+x^3+x^2+x+1 with discriminant 125, which is not a square. The Galois group is generated by (1243), that is an odd permutation. Hence 5 is not in the sequence. n=8: The 8th cyclotomic polynomial is x^4+1 with discriminant 256, which is a square. The Galois group is {id,(13)(57),(15)(37),(17)(35)}, that are all even permutations. Hence 8 is in the sequence.


MAPLE

m := proc(n) local k, r; r := 1;
for k from 1 to n do if igcd(n, k) = 1 then r := modp(r*k, n) fi od; r end:
[1, op(select(n > m(n) = 1, [$1..100]))]; # Peter Luschny, May 25 2017


MATHEMATICA

fQ[n_] := IntegerQ@ Sqrt@ Discriminant[ Cyclotomic[ n, x], x]; Select[ Range@ 100, fQ] (* Robert G. Wilson v, Dec 10 2014 *)


PROG

(PARI) for(n=1, 100, if(issquare(poldisc(polcyclo(n))), print(n)))


CROSSREFS

Cf. A004124, A033949.
Sequence in context: A108059 A276932 A340969 * A139270 A176100 A285658
Adjacent sequences: A180631 A180632 A180633 * A180635 A180636 A180637


KEYWORD

easy,nonn


AUTHOR

Jan Fricke (fricke(AT)mathematik.unisiegen.de), Sep 13 2010


STATUS

approved



