A343003 Numbers k such that there are exactly 2 cyclic cubic fields with discriminant k^2.

63, 91, 117, 133, 171, 217, 247, 259, 279, 301, 333, 387, 403, 427, 469, 481, 511, 549, 553, 559, 589, 603, 657, 679, 703, 711, 721, 763, 793, 817, 871, 873, 889, 927, 949, 973, 981, 1027, 1057, 1099, 1141, 1143, 1147, 1159, 1251, 1261, 1267, 1273, 1333

Offset: 1

Author

Jianing Song, Apr 02 2021

Keywords

• nonn
• easy

Comments

It makes no difference if the word "cyclic" is omitted from the title because a cubic field is cyclic if and only if its discriminant is a square.

Numbers k such that A160498(k) = 4.

Numbers of the form (i) 9p, where p is a prime congruent to 1 modulo 3; (ii) pq, where p, q are distinct primes congruent to 1 modulo 3.

In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is 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), See A343000 for more detailed information.

Examples

			63 is a term since 63^2 = 3969 is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
91 is a term since 91^2 = 8281 is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - x^2 - 30x + 64) and Q[x]/(x^3 - x^2 - 30x - 27).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• LMFDB, Cubic fields
• Wikipedia, Cubic field

Crossrefs

Discriminants and their square roots of cyclic cubic fields:

At least 1 associated cyclic cubic field: A343000, 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, this sequence.

Cf. A006832, A160498, A343023.

Programs

PARI

isA343003(n) = if(omega(n)==2, if(n==63, 1, my(L=factor(n)); L[2,1]%3==1 && L[2,2]==1 && ((L[1,1]%3==1 && L[1,2]==1) || L[1,1]^L[1,2] == 9)), 0)

Formula

a(n) = sqrt(A343002(n)).