This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A308420 #15 Jul 13 2019 16:58:30 %S A308420 2,3,5,6,7,11,13,14,17,21,23,29,33,37,38,47,53,62,69,77,83,93,101,141, %T A308420 167,173,197,213,227,237,293,398,413,437,453,573,677,717,1077,1133, %U A308420 1253,1293,1757 %N A308420 Squarefree numbers d of the form s^2 + r, where r divides 4s, such that Q(sqrt(d)) has class number 1. %C A308420 This sequence is finite, but might not be given in full if the generalized Riemann hypothesis is false. %D A308420 Richard A. Mollin, Quadratics. p. 176, Theorem 5.4.3. Given "a fundamental discriminant of ERD-type with radicand D," the ring of Q(sqrt(D)) has class number 1 "if and only if D" is one of the values listed above, "with one possible exceptional value whose existence would be a counterexample to the GRH" (generalized Riemann hypothesis). %F A308420 Given d = s^2 + r where r | 4s (this is called "extended Richaud-Degert type" or "ERD-type" by Mollin), d is in this sequence if h(O_Q(sqrt(d))) = 1, where h(O_K) is the class number of the ring of algebraic integers O_K. %e A308420 Since 7 = 3^2 - 2 (note that 2 is a divisor of 4 * 9) and h(Z[sqrt(7)]) = 1, 7 is in the sequence. %e A308420 Although 10 = 3^2 + 1, we see that h(Z[sqrt(10)]) > 1 since Z[sqrt(10)] is not a unique factorization domain (e.g., 10 = 2 * 5 = sqrt(10)^2). So 10 is not in the sequence. %e A308420 Although h(Z[sqrt(19)]) = 1, there is no way to express 19 as s^2 + r, e.g., 19 = 3^2 + 10 but 10 is not a divisor of 12, 19 = 4^2 + 3 but 3 is not a divisor of 16, 19 = 5^2 - 6 but 6 is not a divisor of 20, 19 = 6^2 - 17 but -17 does not divide 24. So 19 is not in the sequence either. %Y A308420 Cf. A053329 (first differs at the 16th term), A003172. %K A308420 nonn,fini %O A308420 1,1 %A A308420 _Alonso del Arte_, May 26 2019