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.

10, 79, 82, 226, 730, 1534, 2305, 3601, 4762, 5626, 11026, 21610, 23410, 27226, 38026, 50626, 116554, 164026, 176401, 189226, 342226, 345745, 411394, 518401, 540226, 613090, 804610, 893026, 1071226, 1199026, 1299601, 1334026, 1550026, 2205226, 2433601, 2873026, 3515626, 3920401

Offset: 1

Author

Emmanuel Vantieghem, Mar 13 2017

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

Links

• Robin Visser, Table of n, a(n) for n = 1..50

Crossrefs

Cf. A003172, A003649, A283659.

Programs

Mathematica

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

PARI

classn(n) = qfbclassno(if(n%4>1, 4, 1)*n);
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

Extensions

More terms from Robin Visser, May 25 2024