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A344407
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Discriminant of the (2n)-th cyclotomic field Q(zeta_(2n)). Equivalently, discriminant of the (2n)-th cyclotomic polynomial.
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1
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1, -4, -3, 256, 125, 144, -16807, 16777216, -19683, 4000000, -2357947691, 5308416, 1792160394037, 1157018619904, 1265625, 18446744073709551616, 2862423051509815793, 1586874322944, -5480386857784802185939, 1048576000000000000, 205924456521, 5829995856912430117421056
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OFFSET
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1,2
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COMMENTS
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Note that Q(zeta_n) = Q(zeta_(2n)) for odd n, so this sequence is A004124 with redundant values removed.
a(n) is negative <=> phi(2n) == 2 (mod 4) <=> n = 2 or n is of the form p^e, where p is a prime congruent to 3 modulo 4.
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LINKS
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FORMULA
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For n >= 2, a(p) = (-1)^((p-1)/2) * A130614(n), where p = prime(n).
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EXAMPLE
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n = 2: Q(zeta_4) = Q(i) has discriminant -4;
n = 3: Q(zeta_6) = Q(sqrt(-3)) has discriminant -3;
n = 4: Q(zeta_8) = Q(sqrt(2), i) has discriminant 256;
n = 5: Q(zeta_10) = Q(exp(2*Pi*i/5)) has discriminant 125;
n = 6: Q(zeta_12) = Q(sqrt(3), i) has discriminant 144.
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PROG
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(PARI) vector(25, n, poldisc(polcyclo(2*n)))
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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