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 A343025 Numbers k such that there are at least 2 cyclic cubic fields with discriminant k^2. 7
 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, 819, 871, 873, 889, 927, 949, 973, 981, 1027, 1057, 1099, 1141, 1143, 1147, 1159, 1197, 1251, 1261, 1267 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. Terms in A343001 that are not 9 or a prime. Different from A343002 since a(31) = 819 = 7*9*13. 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. LINKS Jianing Song, Table of n, a(n) for n = 1..1600 LMFDB, Cubic fields Wikipedia, Cubic field FORMULA a(n) = sqrt(A343024(n)). EXAMPLE 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). 819 is a term since 819^2 = 670761 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). PROG (PARI) isA343025(n) = my(L=factor(n), w=omega(n)); 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 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, this sequence. Exactly 2 associated cyclic cubic fields: A343002, A343003. Cf. A006832, A160498, A343023. Sequence in context: A043966 A193417 A062375 * A343003 A253021 A039480 Adjacent sequences:  A343022 A343023 A343024 * A343026 A343027 A343028 KEYWORD nonn,easy AUTHOR Jianing Song, Apr 02 2021 STATUS approved

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Last modified December 2 10:24 EST 2021. Contains 349437 sequences. (Running on oeis4.)