A343002 - OEIS

%I #17 Apr 03 2021 12:13:42

%S 3969,8281,13689,17689,29241,47089,61009,67081,77841,90601,110889,

%T 149769,162409,182329,219961,231361,261121,301401,305809,312481,

%U 346921,363609,431649,461041,494209,505521,519841,582169,628849,667489,758641,762129,790321,859329,900601,946729,962361

%N Discriminants with exactly 2 associated cyclic cubic fields.

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

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

%C 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.

%C Products of two nonequal terms in A343022.

%C 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.

%H Jianing Song, <a href="/A343002/b343002.txt">Table of n, a(n) for n = 1..10000</a>

%H LMFDB, <a href="https://www.lmfdb.org/NumberField/?degree=3">Cubic fields</a>

%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cubic_field">Cubic field</a>

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

%e 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).

%e 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).

%o (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)

%Y Discriminants and their square roots of cyclic cubic fields:

%Y At least 1 associated cyclic cubic field: A343000, A343001.

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

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

%Y Exactly 2 associated cyclic cubic fields: this sequence, A343003.

%Y Cf. A006832, A160498, A343023.

%K nonn,easy

%O 1,1

%A _Jianing Song_, Apr 02 2021