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A038552
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Largest squarefree number k such that Q(sqrt(-k)) has class number n.
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9
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163, 427, 907, 1555, 2683, 3763, 5923, 6307, 10627, 13843, 15667, 17803, 20563, 30067, 34483, 31243, 37123, 48427, 38707, 58507, 61483, 85507, 90787, 111763, 93307, 103027, 103387, 126043, 166147, 134467, 133387, 164803, 222643, 189883
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OFFSET
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1,1
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COMMENTS
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Conjecture: this is also the largest absolute value of negative fundamental discriminant d for class number n. This is to say, for even n, let k be the largest odd number such that h(-k) = n (if it exists), k' be the largest even number such that h(-k') = n (if it exists), then k > k'; here h(D) is the class number of the quadratic field with discriminant D. [Comment rewritten by Jianing Song, Oct 03 2022]
Numbers so far are all 19 mod 24. - Ralf Stephan, Jul 07 2003
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LINKS
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MATHEMATICA
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<< NumberTheory`NumberTheoryFunctions`; a = Table[0, {32} ]; Do[ If[ Mod[n, 4] != 1 || Mod[n, 4] != 2 || SquareFreeQ[n], c = ClassNumber[ -n]; If[c < 33, a[[c]] = n]], {n, 0, 250000} ]; a
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PROG
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(PARI) lista() = {my(nn=10^7, NMAX=100, v = vector(NMAX), c); for (k=1, nn, if (isfundamental(-k), if ((c = qfbclassno(-k)) <= NMAX, v[c]=k); ); ); v; } \\ Michel Marcus, Feb 17 2022; takes several minutes
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CROSSREFS
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KEYWORD
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nonn,nice,hard
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AUTHOR
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Robert Brewer (rbrewerjr(AT)aol.com)
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EXTENSIONS
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2 more terms from Dean Hickerson, May 20 2003. The values were obtained by transcribing and combining data from Tables 1-3 of Buell's paper, which has information for all values of n up to 125.
Values checked against Watkins' data, which proves the values of a(n) for n = 1..100. Charles R Greathouse IV, Feb 08 2012
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STATUS
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approved
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