A343000 Discriminants of cyclic cubic fields.

49, 81, 169, 361, 961, 1369, 1849, 3721, 3969, 4489, 5329, 6241, 8281, 9409, 10609, 11881, 13689, 16129, 17689, 19321, 22801, 24649, 26569, 29241, 32761, 37249, 39601, 44521, 47089, 49729, 52441, 58081, 61009, 67081, 73441, 76729, 77841, 80089, 90601, 94249, 97969

Offset: 1

Author

Jianing Song, Apr 02 2021

Keywords

• nonn
• easy

Comments

Square terms in A006832.

Numbers of the form k^2 where A160498(k) >= 2.

Each term k^2 is associated with A343003(k) cyclic cubic fields.

Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.

Examples

			49 = 7^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 + x + 1).
81 = 9^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3x - 1).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..3200
• LMFDB, Cubic fields
• Wikipedia, Cubic field
• Ka Lun Wong, Maximal Unramified Extensions of Cyclic Cubic Fields, (2011), Theses and Dissertations, 2781.

Crossrefs

Discriminants and their square roots of cyclic cubic fields:

At least 1 associated cyclic cubic field: this sequence, A343001.

Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.

At least 2 associated cyclic cubic fields: A343024, A343025.

Exactly 2 associated cyclic cubic fields: A343002, A343003.

Cf. A006832, A160498, A343023.

Programs

PARI

isA343000(n) = if(issquare(n)&&n>1, my(k=sqrtint(n), L=factor(k), w=omega(k)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 1)

Formula

a(n) = A343001(n)^2.