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A344072
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Smallest even k such that h(-k) = n, where h(D) is the class number of the quadratic field with discriminant D; or 0 if no such k exists.
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4
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20, 56, 104, 164, 296, 356, 404, 584, 1172, 776, 1076, 1316, 1256, 1364, 1844, 1784, 2456, 2504, 4916, 2756, 3176, 3416, 3764, 4424, 4436, 5924, 6296, 4616, 5144, 5444, 10484, 6536, 9236, 7124, 7796, 7556, 12776, 9176, 8564, 10856, 11156, 10436, 11864, 12536, 14804, 13604, 13844, 16376, 15896, 13796
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OFFSET
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1,1
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COMMENTS
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In other words, a(n) is the smallest even k such that Q(sqrt(-k/4)) has class number n; or 0 if no such k exists.
Conjecture 1: a(n) > 0 for all n.
Conjecture 2: If a(n) > 0 and A060649(2n) > 0, then we have a(n) > A060649(2n). This would imply that all terms in A225060 are odd.
Conjecture 3: There exists positive constant c such that a(n) < c*A060649(2n) for all n.
It seems that the ratio a(n)/A060649(2n) reaches its minimum at n = 3. Among the first 250 terms, the maximum of a(n)/A060649(2n) is ~5.3116 attained at n = 227.
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LINKS
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Jianing Song, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Class Number.
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EXAMPLE
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The smallest even k such that h(-k) = 2 is k = 20, so a(1) = 20.
The smallest even k such that h(-k) = 4 is k = 56, so a(2) = 56.
The smallest even k such that h(-k) = 12 is k = 356, so a(6) = 356.
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PROG
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(PARI) a(n) = my(d=4); while(!isfundamental(-d) || qfbclassno(-d)!=2*n, d+=4); d
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CROSSREFS
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Cf. A060649, A225060.
Sequence in context: A144521 A044122 A044503 * A109806 A331774 A216267
Adjacent sequences: A344069 A344070 A344071 * A344073 A344074 A344075
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KEYWORD
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nonn
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AUTHOR
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Jianing Song, May 08 2021
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STATUS
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approved
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