A038552 Largest squarefree number k such that Q(sqrt(-k)) has class number n.

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

Offset

1,1

Comments

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..100

Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796.

Ralf Stephan, Prove or disprove. 100 Conjectures from the OEIS, #16, arXiv:math/0409509 [math.CO], 2004.

M. Watkins, Class numbers of imaginary quadratic fields, Mathematics of Computation 73 (2004), pp. 907-938.

Eric Weisstein's World of Mathematics, Class Number

Mathematica

<< 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

Prog

(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

Crossrefs

Cf. A081319, A046125.

Sequence in context: A142237 A142283 A357600 * A127883 A054466 A221903

Adjacent sequences: A038549 A038550 A038551 * A038553 A038554 A038555

Keyword

nonn,nice,hard

Author

Robert Brewer (rbrewerjr(AT)aol.com)

Extensions

More terms from Robert G. Wilson v, Nov 08 2001

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

Status

approved