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

Author

Robert Brewer (rbrewerjr(AT)aol.com)

Keywords

• nonn
• nice
• hard

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

Crossrefs

Cf. A081319, A046125.

Programs

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

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

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