

A242867


Discriminants of cubic domains for cubefree n.


1



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
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OFFSET

1,2


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)).


REFERENCES

Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 176177, Theorem 7.3.2 on the former page, Table 1 on the latter page.


LINKS

Table of n, a(n) for n=1..41.


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).


EXAMPLE

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.


MATHEMATICA

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


CROSSREFS

Cf. A004709 (cubefree numbers).
Sequence in context: A235292 A202202 A202195 * A255091 A255084 A252337
Adjacent sequences: A242864 A242865 A242866 * A242868 A242869 A242870


KEYWORD

sign


AUTHOR

Alonso del Arte, May 24 2014


STATUS

approved



