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A343002
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Discriminants with exactly 2 associated cyclic cubic fields.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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).
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PROG
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(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)
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CROSSREFS
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Discriminants and their square roots of cyclic cubic fields:
Exactly 2 associated cyclic cubic fields: this sequence, A343003.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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