A344072 Smallest even k such that h(-k) = 2n, where h(D) is the class number of the quadratic field with discriminant D; or 0 if no such k exists.

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

Offset: 1

Author

Jianing Song, May 08 2021

Keywords

• nonn

Comments

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 a 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, which is attained at n = 227.

Examples

			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.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..250
• Eric Weisstein's World of Mathematics, Class Number.

Crossrefs

Cf. A060649, A225060.

Programs

PARI

a(n) = my(d=4); while(!isfundamental(-d) || qfbclassno(-d)!=2*n, d+=4); d