A048941 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 1).

2, 4, 1, 10, 16, 2, 6, 20, 4, 3, 30, 8, 8, 34, 340, 4, 5, 394, 48, 10, 10, 52, 16, 5, 22, 3040, 6, 46, 70, 12, 12, 74, 50, 6, 64, 26, 6964, 20, 7, 48670, 96, 14, 14, 100, 36, 7, 970, 178, 30, 302, 198, 1060, 8, 39, 126, 16, 16, 130, 97684, 8, 25, 502, 6960, 34

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 A346419) give different results, with the first difference at n=14.

(End)

a(n) is the smallest positive integer x 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((u2^2+D*v2^2)/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*u2+D*v1*v2)/q; \\ Jinyuan Wang, Sep 08 2021

Crossrefs

Cf. A000037, A048942, A346419, A346420.

Sequence in context: A135328 A355144 A346419 * A308300 A246188 A135333

Adjacent sequences: A048938 A048939 A048940 * A048942 A048943 A048944

Keyword

nonn

Author

Eric W. Weisstein

Extensions

Name edited by Michel Marcus, Jun 26 2020

Entry revised by Sean A. Irvine, Jul 13 2021

Status

approved