cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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).

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein

Keywords

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

Crossrefs

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

Programs

  • 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

Extensions

Name edited by Michel Marcus, Jun 26 2020
Entry revised by Sean A. Irvine, Jul 16 2021

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).

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein

Keywords

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

Crossrefs

Cf. A000037, A048942, A346419, A346420.

Programs

  • 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

Extensions

Name edited by Michel Marcus, Jun 26 2020
Entry revised by Sean A. Irvine, Jul 13 2021

A346419 a(n) is twice the coefficient of 1 in the fundamental unit of Q(sqrt(A000037(n))) where A000037 lists the nonsquare numbers (Version 2).

Original entry on oeis.org

2, 4, 1, 10, 16, 2, 6, 20, 4, 3, 30, 8, 8, 2, 340, 1, 5, 394, 48, 10, 10, 4, 16, 5, 22, 3040, 2, 46, 70, 12, 12, 74, 50, 6, 64, 26, 6964, 20, 1, 48670, 96, 4, 2, 100, 3, 7, 10, 178, 30, 302, 198, 1060, 8, 39, 126, 16, 16, 130, 97684, 8, 25, 502, 6960, 2, 2136, 86, 4, 340, 9, 106, 160, 1, 18, 164, 5, 9, 20810
Offset: 1

Views

Author

Sean A. Irvine

Keywords

Links

Crossrefs

Cf. A000037, A346420, A048941, A048942.

Programs

  • Mathematica
    nonSquares = Select[Range[72], !IntegerQ[Sqrt[#]]& ]; 2*NumberFieldFundamentalUnits[Sqrt[#]][[1, 2, 1]] & /@ nonSquares (* Jean-François Alcover, Nov 09 2012 *)
  • PARI
    for(n=1,30,if(!issquare(n),print1(abs(2*polcoeff(lift(bnfinit(x^2-n).fu[1]),0)),","))) /* Ralf Stephan, Sep 18 2013; updated by Michel Marcus, Jun 25 2020 */
Showing 1-3 of 3 results.

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