A242867 Discriminants of cubic domains for cubefree n.

1, -108, -243, -108, -675, -972, -1323, -243, -300, -3267, -972, -4563, -5292, -6075, -867, -972, -1083, -2700, -11907, -13068, -14283, -675, -2028, -588, -22707, -24300, -25947, -29403, -31212, -3675, -972, -4107, -38988, -41067, -45387, -47628, -49923, -1452, -6075, -6348, -59643

Offset: 1

Author

Alonso del Arte, May 24 2014

Keywords

• sign

Comments

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)).

Examples

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

References

• Şaban 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.

Crossrefs

Cf. A004709 (cubefree numbers).

Programs

Mathematica

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

Formula

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:

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

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