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%I #18 Dec 31 2021 09:53:07
%S 316,321,469,473,568,817,892,993,1016,1101,1257,1304,1393,1436,1509,
%T 1641,1756,1761,1772,1897,1929,1957,1996,2021,2101,2177,2429,2589,
%U 2636,2908,2913,2981,3173,3261,3356,3569,3736,3873,3928,3941,3957,3981,3997,4009,4193,4281
%N Fundamental discriminants of real quadratic number fields with odd class number > 1 whose fundamental unit has norm 1.
%C Composite terms of A342368.
%C For a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. See Theorem 1 and Theorem 2 of Ezra Brown's link. This sequence gives values for d in the cases (ii) and (iii) and that the real quadratic number field with discriminant d has odd class number > 1.
%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.
%H Henri Cohen and X.-F. Roblot, <a href="http://dx.doi.org/10.1090/S0025-5718-99-01111-4">Computing the Hilbert Class Field of Real Quadratic Fields</a>, Math. Comp. 69 (2000), 1229-1244.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ClassNumber.html">Class Number</a>
%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%e 316 is a term since the quadratic field with discriminant 316 (Q(sqrt(79)) has class number 3. The fundamental unit of that field (80+9*sqrt(79)) has norm 1.
%e 321 is a term since the quadratic field with discriminant 321 (Q(sqrt(321)) has class number 3. The fundamental unit of that field (215+12*sqrt(321)) has norm 1.
%o (PARI) isA349419(D) = if(!isprime(D) && (D>1) && isfundamental(D), my(h=quadclassunit(D)[1]); (h%2)&&(h>1), 0)
%Y Intersection of A342368 and A349649. Equals A342368 \ A350165.
%Y Cf. A003658, A003656, A327297.
%K nonn
%O 1,1
%A _Jianing Song_, Dec 29 2021
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