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

A079941 Greedy frac multiples of log(2): a(1)=1, Sum_{n>0} frac(a(n)*log(2)) = 1.

Original entry on oeis.org

1, 3, 6, 13, 26, 39, 277, 642, 2291, 4582, 6231, 16402, 26573, 36744, 63317, 73488, 110232, 414355, 828710, 1206321, 2412642, 4410929, 5617250, 12026466, 31668469, 51310472, 70952475, 141904950, 394046381
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(3x) + frac(6x) + frac(13x) < 1, while frac(1x) + frac(3x) + frac(6x) + frac(k*x) > 1 for all k>6 and k<13.

Crossrefs

Cf. A079943 (denominators of convergents to ln2), A079934, A079939, A079940.

Extensions

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003
a(20)-a(29) from Sean A. Irvine, Aug 31 2025

A080142 Greedy frac multiples of 1/Pi: a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=1/Pi, where "frac(y)" denotes the fractional part of y.

Original entry on oeis.org

1, 2, 22, 44, 66, 88, 110, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 3550, 3905, 4260, 4615, 4970, 5325, 5680, 6035, 6390, 6745, 7100, 7455, 7810, 8165, 104348, 104703, 105058, 105413, 105768, 208696, 209051, 312689, 313044, 417037
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 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(3) = 22 since frac(1x) + frac(2x) + frac(22x) < 1, while frac(1x) + frac(2x) + frac(k*x) > 1 for all k>2 and k<22.

Crossrefs

Cf. A079938, A079939, A079940, A079941, etc.

Programs

  • Maple
    Digits := 1000: a := []: s := 0: x := evalf(1.0/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;
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, fps = Plus @@ Table[FractionalPart[a[i]*Pi^-1], {i, n - 1}]}, While[fps + FractionalPart[k*Pi^-1] > 1, k++ ]; a[n] = k]; Do[ Print[ a[n]], {n, 40}] (* Robert G. Wilson v, Nov 03 2004 *)

A080157 Greedy frac multiples of gamma: a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=gamma, where "frac(y)" denotes the fractional part of y.

Original entry on oeis.org

1, 2, 7, 9, 26, 52, 149, 272, 395, 790, 1185, 1580, 5653, 10911, 16169, 26685, 58628, 85313, 117256, 175884, 559595, 2179752, 5420066
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

Keywords

Examples

			a(3) = 7 since frac(1x) + frac(2x) + frac(7x) < 1, while frac(1x) + frac(2x) + frac(k*x) > 1 for all k>2 and k<7.

Crossrefs

Cf. A079938, A079939, A079940, A079941, A080142. Searching in the OEIS for "greedy frac" gives related sequences.

Programs

  • Maple
    Digits := 1000: a := []: s := 0: x := evalf(gamma): 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;

A080158 Greedy frac multiples of Catalan's constant, G: a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=G, where "frac(y)" denotes the fractional part of y.

Original entry on oeis.org

1, 11, 107, 10579, 21158, 53014, 106028, 625708, 721157, 1442314, 2163471, 2884628, 3605785, 4326942
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

Keywords

Comments

For definition of how the "Greedy Frac" sequence is defined, see other sequences in index.

Examples

			a(3) = 107 since frac(1x) + frac(11x) + frac(107x) < 1, while frac(1x) + frac(11x) + frac(k*x) > 1 for all k>11 and k<107.

Crossrefs

Cf. A079938, A079939, A079940, A079941, A080142, A080157.

Programs

  • Maple
    Digits := 1000: a := []: s := 0: x := evalf(Catalan): for n from 1 to 5000000 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;
Showing 1-5 of 5 results.

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