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A005472 Class numbers of Shanks' simplest cubic fields.

%I A005472 M3291 #36 Dec 08 2024 17:27:59
%S A005472 1,1,1,1,1,1,1,4,7,4,4,4,7,4,13,7,19,7,7,7,19,19,19,16,31,19,28,19,49,
%T A005472 31,28,31,64,43,37,127,61,52,52,52,49,100,37,112,64,67,61,76,61,76,61,
%U A005472 61,112,76,73,67,133,91,223,169,73,112,100,169,91,121,175
%N A005472 Class numbers of Shanks' simplest cubic fields.
%C A005472 Class numbers of cubic fields with discriminants p^2, where p runs through the primes in A005471.
%C A005472 All terms are of the form x^2 + 3*y^2 (A003136). - _Colin Barker_, Nov 30 2014
%D A005472 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005472 Robin Visser, <a href="/A005472/b005472.txt">Table of n, a(n) for n = 1..10000</a> (terms n = 1..100 from R. J. Mathar).
%H A005472 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8">The simplest cubic fields</a>, Math. Comp., 28 (1974), 1137-1152 (see Table 1 page 1140).
%o A005472 (PARI) A175282(n)={
%o A005472     local(a);
%o A005472     if(n==1,
%o A005472         return(1),
%o A005472         a=A175282(n-1)+1;
%o A005472         while(1,
%o A005472             if( isprime(a^2+3*a+9),
%o A005472                 return(a),
%o A005472                 a++
%o A005472             );
%o A005472         )
%o A005472     )
%o A005472 };
%o A005472 A005472(n)={
%o A005472     local(a,bnf,L,H);
%o A005472     if(n==1, return(1));
%o A005472     a=A175282(n);
%o A005472     bnf=bnfinit(x^3-a*x^2-(a+3)*x-1);
%o A005472     L=ideallist(bnf,1,2);
%o A005472     H=bnrclassnolist(bnf,L);
%o A005472     return(H[1][1]);
%o A005472 };
%o A005472 for(n=1,80, print1(A005472(n)," ") ); /* _R. J. Mathar_, Jun 06 2019 */
%Y A005472 Cf. A003136, A005471, A005474.
%K A005472 nonn
%O A005472 1,8
%A A005472 _N. J. A. Sloane_
%E A005472 Name edited by _Robin Visser_, Dec 06 2024