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 1-5 of 5 results.

A088900 Numbers n such that 2*n*k(n) is an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 55, 56, 64, 70, 72, 81, 90, 100, 110, 121, 132, 144, 155, 156, 169, 180, 182, 196, 210, 225, 240, 256, 272, 289, 305, 306, 324, 342, 361, 377, 380, 400, 420, 441, 462, 484, 504, 505, 506, 529, 546, 552, 576, 600, 625
Offset: 1

Views

Author

Thomas Baruchel, Oct 21 2003

Keywords

Comments

Subset of A086378.

Crossrefs

Cf. A086378.

A087414 Numbers n such that 2*n*k(n) is rational but not an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there); numbers belonging to A086378 but not to A088900.

Original entry on oeis.org

153, 1717, 2244, 2340, 3525, 3650, 6460, 7119, 7475, 10074, 14490, 19147, 20008, 20862, 21424, 21747, 24453, 25400, 26039, 27346, 28028, 28371, 31484, 35483, 37008, 44275, 44678, 45974, 50389, 52155, 62187, 63724, 64752
Offset: 1

Views

Author

Thomas Baruchel, Oct 21 2003

Keywords

Crossrefs

Cf. A086378 and A088900.

Programs

  • PARI
    /* z(n)!=0 iff n is in the sequence */
z(n)= { local(a,b,c,d,e,f,g,h,i,j,k);
b=a=sqrtint(n);d=f=i=1;e=g=h=0;j=c=n-a^2;if(!c,return(0));
until((a==b)&&(c==j),k=d+a*e;f*=c;d=a*d+e*n;e=k;g+=i;i*=c;
k=g+a*h;g=a*g+h*n;h=k;k=(a+b)\c;g-=i*k;a=c*k-a;c=(n-a^2)/c);
d=d/f-1;e/=f;g/=i;h/=i;i=d^2-n*e^2;k=h*d-g*e;g=g*d-h*e*n;
b=n-a^2;a=b*g-c*a*i;c=b*k+i*c;b*=i;!a*(2%(b/gcd(b,n*c))); }

A088901 Numbers n such that n*k(n) is an integer, where k(n) is sum of successive remainders when computing the Euclidean algorithm for (1, 1/sqrt(n)) as defined in A086378 (MuPAD program is given there).

Original entry on oeis.org

1, 2, 4, 9, 12, 16, 25, 30, 36, 49, 55, 56, 64, 70, 81, 90, 100, 121, 132, 144, 169, 180, 182, 196, 225, 240, 256, 289, 305, 306, 324, 361, 377, 380, 400, 441, 462, 484, 529, 552, 576, 625, 646, 650, 676, 729, 755, 756, 784, 841, 870, 900, 961, 987, 990, 992
Offset: 1

Views

Author

Thomas Baruchel, Oct 21 2003

Keywords

Comments

Subset of A086378 and A088900.

Crossrefs

Cf. A086378 and A088900.

A087947 Sum of successive remainders in computing Euclidean algorithm for (1, 1/sqrt(-n)) has real and imaginary parts equal.

Original entry on oeis.org

1, 4, 5, 6, 8, 9, 16, 17, 18, 20, 24, 25, 36, 37, 38, 39, 40, 42, 48, 49, 64, 65, 66, 68, 72, 78, 80, 81, 100, 101, 102, 104, 105, 110, 117, 120, 121, 144, 145, 146, 147, 148, 150, 152, 155, 156, 164, 168, 169, 196, 197, 198, 200, 203, 210, 220, 222, 224, 225, 256
Offset: 1

Views

Author

Thomas Baruchel, Sep 07 2003

Keywords

Comments

Since the computation of the algorithm needs an extension of the integer part over a subset of C, the rule: floor(I*x) = i*floor(x) is used (which is what MuPAD does). The following program computes the exact value of the sum.
For all a(n) in the sequence, the relation: (2k)^2 <= a(n) <= (2k+1)^2 is true.

Examples

			kappa(1/sqrt(-203)) = (1/2 + (1/2)i) - (1/29 + (1/29)i)*sqrt(203).

Crossrefs

Cf. A086378, A087948.

Programs

  • MuPAD
    kappa_1_over_comp_sqrt := proc(n) local a,b,i,p; begin if (a := sqrt(-n)-isqrt(-n)) = 0 then return(0) end_if: a := simplify(1/a,sqrt); i := a := simplify(1/(a - floor(a)),sqrt); p := 1; b := 0; repeat p := p*a; b := b*a+a-floor(a); until (a := simplify(1/(a-floor(a)),sqrt)) = i end_repeat: return(simplify((1-isqrt(n)/sqrt(n))*(1+b/(p-1)+1/a-floor(1/a)),sqrt)); end_proc:

A087948 Sum of successive remainders in computing euclidean algorithm for (1, -1/sqrt(-n)) has real and imaginary parts equal.

Original entry on oeis.org

1, 4, 5, 9, 16, 17, 18, 25, 36, 37, 39, 49, 64, 65, 66, 68, 81, 100, 101, 105, 121, 126, 144, 145, 146, 147, 150, 169, 196, 197, 203, 225, 256, 257, 258, 260, 264, 289, 324, 325, 327, 333, 361, 400, 401, 402, 405, 410, 441, 484, 485, 495, 529, 576, 577, 578, 579
Offset: 1

Views

Author

Thomas Baruchel, Sep 07 2003

Keywords

Comments

Since the computation of the algorithm needs an extension of the integer part over a subset of C, the rule: floor(i*x) = i*floor(x) is used (which is what MuPAD does). The following program computes the exact value of the sum.

Examples

			kappa(-1/sqrt(-105)) = -(1/210 + (1/210)i)*sqrt(105).

Crossrefs

Cf. A086378, A087947.

Programs

  • MuPAD
    kappa_neg_1_over_comp_sqrt := proc(n) local a,b,i,p; begin if (a := -sqrt(-n)+ceil(sqrt(-n))) = 0 then return(0) end_if: i := a := simplify(1/a,sqrt); p := 1; b := 0; repeat p := p*a; b := b*a+a-floor(a); until (a := simplify(1/(a-floor(a)),sqrt)) = i end_repeat: return(simplify(-(b/(p-1) + 1/a)/sqrt(-n),sqrt)); end_proc:
Showing 1-5 of 5 results.

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