A390079 - OEIS

%I #72 Jun 11 2026 13:46:20

%S 5,8,12,13,17,20,21,24,28,29,32,33,37,40,41,44,45,48,52,53,56,57,60,

%T 61,65,68,69,72,73,76,77,80,84,85,88,89,92,93,96,97,101,104,105,108,

%U 109,112,113,116,117,120,124,125,128,129,132,133,137,140,141,149,152,153,156,157,160,161,164,165,168,172,173,176,177,180,181,184,185,188,189,192,193,197,200

%N Discriminants of orders of real quadratic fields with 1 class per genus.

%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. Sequence gives D such that Cl+(D) = (C_2)^r for some r >= 0.

%C A391419 gives D such that Cl(D) = (C_2)^r. For D in A391419:

%C  - If the fundamental unit has norm -1, then Cl+(D) = Cl(D), so D is in this sequence;

%C  - If 16|D or D has a prime factor congruent to 3 modulo 4, then Cl+(D) = Cl(D) X C_2, so D is in this sequence;

%C  - In the remaining cases, Cl+(D) has a cyclic subgroup of order 4, so D is not in this sequence.

%C See A391439 for more details.

%H Jianing Song, <a href="/A390079/b390079.txt">Table of n, a(n) for n = 1..10000</a>

%H Rick L. Shepherd, <a href="https://www.digitalgreensboro.org/record/14435">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

%o (PARI) isA390079(n) = if(n%4<=1 && !issquare(n) && !#select(k->k<>2, quadclassunit(n).cyc), if(n%16==0 || quadunitnorm(n)==-1, return(1)); my(f = factor(n)[,1]~); for(i=1, #f, if(f[i]%4==3, return(1)))); return(0) \\ quadunitnorm() requires PARI-GP of version 2.15 or higher

%Y Cf. A003171 (the sequence for imaginary quadratic fields).

%Y Cf. A306638 (norms of the fundamental unit).

%Y Sequences related to the class groups of real quadratic fields:

%Y For class groups related to fundamental discriminants (A003658): A391436, A391437 (2-rank), A391426, A391435 (number of genera), A391417 (exponent <= 2);

%Y For form class groups related to fundamental discriminants: A317991, A317992 (2-rank), A317989, A317990 (number of genera), A391422 (exponent <= 2);

%Y For class groups related to all discriminants (A079896): A391439 (2-rank), A391438 (number of genera), A391419 (exponent <= 2);

%Y For form class groups related to all discriminants: A391441 (2-rank), A391440 (number of genera), this sequence (exponent <= 2).

%K nonn,changed

%O 1,1

%A _Jianing Song_, Nov 24 2025