A343002 Discriminants with exactly 2 associated cyclic cubic fields.

3969, 8281, 13689, 17689, 29241, 47089, 61009, 67081, 77841, 90601, 110889, 149769, 162409, 182329, 219961, 231361, 261121, 301401, 305809, 312481, 346921, 363609, 431649, 461041, 494209, 505521, 519841, 582169, 628849, 667489, 758641, 762129, 790321, 859329, 900601, 946729, 962361

Offset: 1

Author

Jianing Song, Apr 02 2021

Keywords

• nonn
• easy

Comments

A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.

Numbers of the form k^2 where A160498(k) = 4.

Numbers of the form k^2 where k is 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.

Products of two nonequal terms in A343022.

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

			3969 = 63^2 is a term since it is the discriminant of the 2 cyclic cubic fields Q[x]/(x^3 - 21x - 28) and Q[x]/(x^3 - 21x - 35).
8281 = 91^2 is a term since it 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: this sequence, A343003.

Cf. A006832, A160498, A343023.

Programs

PARI

isA343002(n) = if(omega(n)==2, if(n==3969, 1, my(L=factor(n)); L[2,1]%3==1 && L[2,2]==2 && ((L[1,1]%3==1 && L[1,2]==2) || L[1,1]^L[1,2] == 81)), 0)

Formula

a(n) = A343003(n)^2.