A391531 - OEIS

%I #12 Dec 14 2025 08:03:37

%S 60,105,120,140,156,165,168,204,220,264,273,280,285,312,345,348,357,

%T 364,380,385,408,429,440,444,456,460,465,476,492,552,561,572,609,616,

%U 620,636,645,665,696,705,728,732,741,744,748,760,805,812,860,861,885,888,920,952,957,969,984,988

%N Discriminants of real quadratic fields with class group C_2 and form class group C_2 X C_2.

%C Let Cl+(D) and Cl(D) be the narrow class group and the class group of the quadratic order of discriminant D. Then Cl+(D)/Cl(D) = 1 if D < 0 or D > 0 and the fundamental unit has norm -1, C_2 if D > 0 and the fundamental unit has norm 1.

%C For fundamental discriminants D, the 2-rank of Cl+(D) is omega(D) - 1, omega = A001221. Hence

%C   Cl(D) = C_2 and Cl+(D) = C_2 X C_2 <=> Cl(D) = C_2 and omega(D) = 3.

%C Each term has a prime factor congruent to 3 modulo 4; see A391439 for a proof.

%H Jianing Song, <a href="/A391531/b391531.txt">Table of n, a(n) for n = 1..10000</a>

%H Ezra Brown, <a href="https://doi.org/10.1090/S0002-9947-1974-0364172-9">Class numbers of real quadratic number fields</a>, Trans. Amer. Math. Soc. 190 (1974), 99-107.

%o (PARI) isA391531(D) = D>1 && isfundamental(D) && qfbclassno(D)==2 && omega(D)==3

%Y Fundamental discriminants D of real quadratic fields such that

%Y Cl(D) trivial, Cl+(D) trivial: A003656;

%Y Cl(D) trivial, Cl+(D) = C_2: A327297;

%Y Cl(D) = C_2, Cl+(D) = C_2: A391529;

%Y Cl(D) = C_2, Cl+(D) = C_4: A391530;

%Y Cl(D) = C_2, Cl+(D) = C_2 X C_2: this sequence;

%Y Cl(D) = C_3, Cl+(D) = C_3: A391421;

%Y Cl(D) = C_3, Cl+(D) = C_6: A391532.

%K nonn

%O 1,1

%A _Jianing Song_, Dec 12 2025