A048942 a(n) is twice the coefficient of the radical part in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

2, 2, 1, 4, 6, 1, 2, 6, 1, 1, 8, 2, 2, 8, 78, 1, 1, 84, 10, 2, 2, 10, 3, 1, 4, 546, 1, 8, 12, 2, 2, 12, 8, 1, 10, 4, 1062, 3, 1, 7176, 14, 2, 2, 14, 5, 1, 132, 24, 4, 40, 26, 138, 1, 5, 16, 2, 2, 16, 11934, 1, 3, 60, 826, 4, 250, 10, 6, 39, 1, 12, 18, 2, 2, 18

Offset

1,1

Comments

From Sean A. Irvine, Jul 16 2021: (Start)

These values are computed by Algorithm 5.7.2 in Cohen.

Other methods of computation (see A346420) give different results, with the first difference at n=14.

(End)

a(n) is the smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = +-4, where D = A000037(n). - Jinyuan Wang, Sep 08 2021

References

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993.

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

S. R. Finch, Class number theory

Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Sean A. Irvine, Java program (github)

Eric Weisstein's World of Mathematics, Fundamental Unit.

Prog

(PARI) a(n) = my(A, D=n+(1+sqrtint(4*n))\2, d=sqrtint(D), p, q, t, u1, u2, v1, v2); if(d%2==D%2, p=d, p=d-1); u1=-p; u2=2; v1=1; v2=0; q=2; while(v2==0 || q!=t, A=(p+d)\q; t=p; p=A*q-p; if(t==p && v2!=0, return(2*u2*v2/q), t=A*u2+u1; u1=u2; u2=t; t=A*v2+v1; v1=v2; v2=t; t=q; q=(D-p^2)/q)); (u1*v2+u2*v1)/q; \\ Jinyuan Wang, Sep 08 2021

Crossrefs

Cf. A000037, A007913, A048941, A346419, A346420.

Sequence in context: A330664 A330843 A115313 * A121484 A273903 A346420

Adjacent sequences: A048939 A048940 A048941 * A048943 A048944 A048945

Keyword

nonn

Author

Eric W. Weisstein

Extensions

Name edited by Michel Marcus, Jun 26 2020

Entry revised by Sean A. Irvine, Jul 16 2021

Status

approved