A005472 Class numbers of Shanks' simplest cubic fields. (Formerly M3291)

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

1,8

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).

Prog

(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 */

Crossrefs

Cf. A003136, A005471, A005474.

Sequence in context: A020803 A019626 A203137 * A111522 A342203 A376549

Adjacent sequences: A005469 A005470 A005471 * A005473 A005474 A005475

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Name edited by Robin Visser, Dec 06 2024

Status

approved