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A242867 Discriminants of cubic domains for cubefree n. 1

%I #19 Jun 12 2014 13:22:47

%S 1,-108,-243,-108,-675,-972,-1323,-243,-300,-3267,-972,-4563,-5292,

%T -6075,-867,-972,-1083,-2700,-11907,-13068,-14283,-675,-2028,-588,

%U -22707,-24300,-25947,-29403,-31212,-3675,-972,-4107,-38988,-41067,-45387,-47628,-49923,-1452,-6075,-6348,-59643

%N Discriminants of cubic domains for cubefree n.

%C The table in Alaca & Williams (2004) skips over n = 4, 9, 16 but includes 12, 18 and 20; then there is a footnote to the table explaining that Q(4^(1/3)) and Q(16^(1/3)) work out to be subdomains of Q(2^(1/3)), and similarly for Q(9^(1/3)) and Q(3^(1/3)) and for Q(18^(1/3)) and Q(12^(1/3)).

%D ┼×aban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 176-177, Theorem 7.3.2 on the former page, Table 1 on the latter page.

%F Set m = A004709(n), then express it as m = h * k^2, where k = A000188(m), the square root of the largest square dividing m, and h = m/k^2 = A007913(m). Then:

%F a(n) = -3h^2 * k^2 if m == +-1 (mod 9), otherwise a(n) = -27h^2 * k^2.

%F This formula is from Theorem 7.3.2 in Alaca & Williams (2004).

%e a(7) = -1323 because the seventh cubefree number is 7 and Q(7^(1/3)) has -1323 for a discriminant.

%e a(8) = -243 because the eighth cubefree number is 9 and Q(9^(1/3)) is a subdomain of Q(3^(1/3)), which has a discriminant of -243.

%t DeleteCases[Table[Boole[FreeQ[FactorInteger[n], {_, k_ /; k > 2}]] * NumberFieldDiscriminant[n^(1/3)], {n, 100}], 0]

%Y Cf. A004709 (cubefree numbers).

%K sign

%O 1,2

%A _Alonso del Arte_, May 24 2014

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Last modified March 4 23:31 EST 2024. Contains 370537 sequences. (Running on oeis4.)