A343024 Discriminants with at least 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, 670761, 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.

Terms in A343000 that are not 81 or a square of a prime.

Different from A343002 since a(31) = 819^2 = (7*9*13)^2.

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

			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).
670761 = 819^2 is a term since it is the discriminant of the 4 cyclic cubic fields Q[x]/(x^3 - 273x - 91), Q[x]/(x^3 - 273x - 728), Q[x]/(x^3 - 273x - 1547) and Q[x]/(x^3 - 273x - 1729).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..1600
• 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: this sequence, A343025.

Exactly 2 associated cyclic cubic fields: A343002, A343003.

Cf. A006832, A160498, A343023.

Programs

PARI

isA343024(n) = if(issquare(n), my(k=sqrtint(n), L=factor(k), w=omega(k)); if(w<2, return(0)); 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) = A343025(n)^2.