A005472 Class numbers of Shanks' simplest cubic fields.

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, 31, 28, 31, 64, 43, 37, 127, 61, 52, 52, 52, 49, 100, 37, 112, 64, 67, 61, 76, 61, 76, 61, 61, 112, 76, 73, 67, 133, 91, 223, 169, 73, 112, 100, 169, 91, 121, 175

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Class numbers of cubic fields with discriminants p^2, where p runs through the primes in A005471.

All terms are of the form x^2 + 3*y^2 (A003136). - Colin Barker, Nov 30 2014

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..100 from R. J. Mathar).
• D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152 (see Table 1 page 1140).

Crossrefs

Cf. A003136, A005471, A005474.

Programs

PARI

A175282(n)={
    local(a);
    if(n==1,
        return(1),
        a=A175282(n-1)+1;
        while(1,
            if( isprime(a^2+3*a+9),
                return(a),
                a++
            );
        )
    )
};
A005472(n)={
    local(a,bnf,L,H);
    if(n==1, return(1));
    a=A175282(n);
    bnf=bnfinit(x^3-a*x^2-(a+3)*x-1);
    L=ideallist(bnf,1,2);
    H=bnrclassnolist(bnf,L);
    return(H[1][1]);
};
for(n=1,80, print1(A005472(n)," ") ); /* R. J. Mathar, Jun 06 2019 */

Extensions

Name edited by Robin Visser, Dec 06 2024