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A283658 Numbers d > 1 such that the class number of Q(sqrt(d)) is strictly greater than the class number of Q(sqrt(m)) for all m < d.

%I A283658 #21 May 26 2024 18:00:07
%S A283658 10,79,82,226,730,1534,2305,3601,4762,5626,11026,21610,23410,27226,
%T A283658 38026,50626,116554,164026,176401,189226,342226,345745,411394,518401,
%U A283658 540226,613090,804610,893026,1071226,1199026,1299601,1334026,1550026,2205226,2433601,2873026,3515626,3920401
%N A283658 Numbers d > 1 such that the class number of Q(sqrt(d)) is strictly greater than the class number of Q(sqrt(m)) for all m < d.
%C A283658 Every element d of the sequence is squarefree because, if f is the squarefree part of d, then Q(sqrt(f)) = Q(sqrt(d)). If f would be < d, the class number of Q(sqrt(f)) would not be < the class number of Q(sqrt(d)). Thus, f = d.
%D A283658 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%H A283658 Robin Visser, <a href="/A283658/b283658.txt">Table of n, a(n) for n = 1..50</a>
%e A283658 The sequence starts with 10 because the class number of Q(sqrt(10)) = 2 and all fields Q(sqrt(m)) with m < 10 have class number 1.
%e A283658 The next term is 79 because the class number of Q(sqrt(79)) is 3 and all fields Q(sqrt(m)) with m < 79 have class number 1 or 2.
%t A283658 A={}; hx = 1; d = 2; While[hx<300, d++; If[SquareFreeQ[d], h = NumberFieldClassNumber[Sqrt[d]]; If[h > hx, AppendTo[A,d]; hx = h]]]; A
%o A283658 (PARI) classn(n) = qfbclassno(if(n%4>1, 4, 1)*n);
%o A283658 isok(d) = {if (issquarefree(d), cld = classn(d); for (k=2, d-1, if (issquarefree(k) && (classn(k) >= cld), return (0))); 1;);} \\ _Michel Marcus_, Mar 13 2017
%Y A283658 Cf. A003172, A003649, A283659.
%K A283658 nonn
%O A283658 1,1
%A A283658 _Emmanuel Vantieghem_, Mar 13 2017
%E A283658 More terms from _Robin Visser_, May 25 2024