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A343000
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Discriminants of cyclic cubic fields.
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8
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49, 81, 169, 361, 961, 1369, 1849, 3721, 3969, 4489, 5329, 6241, 8281, 9409, 10609, 11881, 13689, 16129, 17689, 19321, 22801, 24649, 26569, 29241, 32761, 37249, 39601, 44521, 47089, 49729, 52441, 58081, 61009, 67081, 73441, 76729, 77841, 80089, 90601, 94249, 97969
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OFFSET
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1,1
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COMMENTS
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Numbers of the form k^2 where A160498(k) >= 2.
Each term k^2 is associated with A343003(k) cyclic cubic fields.
Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be 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), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.
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LINKS
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FORMULA
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EXAMPLE
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49 = 7^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 + x + 1).
81 = 9^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3x - 1).
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PROG
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(PARI) isA343000(n) = if(issquare(n)&&n>1, my(k=sqrtint(n), L=factor(k), w=omega(k)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 1)
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CROSSREFS
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Discriminants and their square roots of cyclic cubic fields:
At least 1 associated cyclic cubic field: this sequence, A343001.
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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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