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.

A079939 Greedy frac multiples of e: a(1)=1, Sum_{n>0} frac(a(n)*e)=1.

Original entry on oeis.org

1, 3, 7, 14, 39, 78, 394, 1001, 2002, 3003, 9545, 10546, 27634, 154257, 398959, 797918, 1196877, 1595836, 5394991, 5793950, 15786014, 130087267, 312129649, 624259298
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

Keywords

Comments

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

Examples

			a(4) = 14 since frac(1x) + frac(3x) + frac(7x) + frac(14x) < 1, while frac(1x) + frac(3x) + frac(7x) + frac(k*x) > 1 for all k>7 and k<14.

Crossrefs

Cf. A007677 (denominators of convergents to e), A079934, A079937, A079940.

Programs

  • Maple
    Digits := 100: a := []: s := 0: x := exp(1.0): for n from 1 to 1000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

Extensions

a(15)-a(16) from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003
a(17)-a(24) from Sean A. Irvine, Aug 30 2025

A079938 Greedy frac multiples of Pi: a(1)=1, Sum_{n>=1} frac(a(n)*Pi) = 1.

Original entry on oeis.org

1, 2, 3, 8, 99, 33102, 66317, 265381, 1360120, 25510582, 78256779, 156513558, 209259755, 340262731, 1963319607, 6701487259, 8664806866, 13402974518, 20104461777, 26805949036, 33507436295, 40208923554, 46910410813
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

Keywords

Comments

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

Examples

			a(4) = 8 since frac(1x*) + frac(2*x) + frac(3*x) + frac(8*x) < 1, while frac(1*x) + frac(2*x) + frac(3*x) + frac(k*x) > 1 for all k > 3 and k < 8.

Crossrefs

Cf. A002486 (denominators of convergents to Pi), A079934, A079937, A079939.

Programs

  • Maple
    Digits := 100: a := []: s := 0: x := Pi: for n from 1 to 10000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;
  • PARI
    first(n)=my(v=vector(n),s=1.,p=Pi-3,k); for(m=1,oo, my(t=frac(p*m)); if(tCharles R Greathouse IV, Jul 25 2024

Extensions

a(9) from Mark Hudson, Jan 30 2003
a(10)-a(23) from Charles R Greathouse IV, Jul 26 2024

A079936 Greedy frac multiples of sqrt(5): a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=sqrt(5).

Original entry on oeis.org

1, 2, 5, 13, 17, 34, 305, 610, 1597, 4181, 5473, 10946, 98209, 196418, 514229, 1346269, 1762289, 3524578, 31622993, 63245986, 165580141, 433494437, 567451585, 1134903170, 10182505537, 20365011074, 53316291173, 139583862445
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

Keywords

Comments

The n-th greedy frac multiple of x is the smallest integer that does not cause sum(k=1..n,frac(a(k)*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

Examples

			a(4) = 13 since frac(1x) + frac(2x) + frac(5x) + frac(13x) < 1, while frac(1x) + frac(2x) + frac(5x) + frac(k*x) > 1 for all k>5 and k<13.

Links

Crossrefs

Cf. A001076 (denominators of convergents to sqrt(5)), A079934, A079935, A079937.

Formula

For n>=0, a(6n+1)=A001076(4n+1); a(6n+2)=2a(6n+1); a(6n+3)=A001076(4n+1)+A001076(4n+2); a(6n+4)=A001076(4n+3)-A001076(4n+2); a(6n+5)=A001076(4n+3); a(6n+6)=2a(6n+5). Asymptotics: a(6n) -> 2*sqrt(5)*(tau)^(12n-3); a(6n+2)/a(6n+1) -> (tau)^2; a(6n+3)/a(6n+2) -> (tau)^2; a(6n+4)/a(6n+3) -> (tau)^2/2; a(6n+6)/a(6n+5) -> (tau)^6/2; where tau = (1+sqrt(5))/2.
G.f.: -x*(x -1)*(2*x^10 +3*x^9 +8*x^8 +21*x^7 +55*x^6 +72*x^5 +38*x^4 +21*x^3 +8*x^2 +3*x +1) / (x^12 -322*x^6 +1). - Colin Barker, Jun 16 2013

A080017 Denominators of the convergents to the continued fraction of Pi^2/6.

Original entry on oeis.org

1, 1, 2, 3, 14, 31, 138, 997, 1135, 5537, 12209, 42164, 180865, 1850814, 2031679, 5914172, 7945851, 13860023, 21805874, 340948133, 362754007, 1429210154, 8938014931, 10367225085, 19305240016, 48977705117, 68282945133, 117260650250, 185543595383, 488347841016
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2003

Keywords

Crossrefs

Cf. A079937, A080016 (numerators).

Programs

  • Mathematica
    Denominator[Convergents[Pi^2/6,30]] (* Harvey P. Dale, Jul 08 2017 *)

Extensions

More terms from Harvey P. Dale, Jul 08 2017

A080016 Numerators of the convergents to the continued fraction of Pi^2/6.

Original entry on oeis.org

1, 2, 3, 5, 23, 51, 227, 1640, 1867, 9108, 20083, 69357, 297511, 3044467, 3341978, 9728423, 13070401, 22798824, 35869225, 560837199, 596706424, 2350956471, 14702445250
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2003

Keywords

Crossrefs

Cf. A079937, A080017 (denominators).
Showing 1-5 of 5 results.

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