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A180634
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Numbers n such that the discriminant of the n-th cyclotomic polynomial is a square.
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1
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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
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1,2
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COMMENTS
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A number n is in this sequence if the Galois group of the n-th cyclotomic polynomial over the rationals contains only even permutations.
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
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LINKS
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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. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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fQ[n_] := IntegerQ@ Sqrt@ Discriminant[ Cyclotomic[ n, x], x]; Select[ Range@ 100, fQ] (* Robert G. Wilson v, Dec 10 2014 *)
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PROG
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(PARI) for(n=1, 100, if(issquare(poldisc(polcyclo(n))), print(n)))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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