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.

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A227773 Least splitter of f(n) and f(n+1), where s(1) = 1, s(2) = 1, s(n) = s(n-1) + s(n-2)/(n-2) and f(n) = n/(n - s(n)).

Original entry on oeis.org

1, 3, 3, 18, 39, 71, 323, 536, 1001, 8544, 45723, 208524, 398959, 3400196, 5394991, 10391023, 150869313, 1097649283, 5467464369, 10622799089, 132941053437, 403978495031, 403978495031, 8286870547680, 76601727404275, 399178399621704, 781379079653017
Offset: 2

Views

Author

Clark Kimberling, Jul 30 2013

Keywords

Comments

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Since f(n) -> e, the corresponding least splitting rationals (see Example) also approach e; e.g., |f(30) - e| < 10^(-33).

Examples

			The least splitters are the denominators of the least splitting rationals for f(n) and f(n+1):
2/1, 8/3, 8/3, 49/18, 106/39, 193/71, 878/323, 1457/536, 2721/1001, 23225/8544, 124288/45723, 566827/208524, 1084483/398959, 9242691/3400196, 14665106/5394991, 28245729/10391023
f(2) = 2 <= 2/1 < f(3) = 3;
f(3) = 3 > 8/3 >= f(4) = 8/3;
f(4) = 8/3 <= 8/3 < f(5) = 30/11;
f(5) = 30/11 > 49/18 >= f(6) = 144/53.

Links

Crossrefs

Cf. A227631, A227688.

Programs

  • Mathematica
    z = 17; r[x_, y_] := Module[{a, b, x1 = Min[{x, y}], y1 = Max[{x, y}]}, If[x == y, x, b = NestWhile[#1 + 1 &, 1, ! (a = Ceiling[#1 x1 - 1]) < Ceiling[#1 y1] - 1 &]; (a + 1)/    b]]; s[1] = 1; s[2] = 1; s[n_] := s[n] = s[n - 1] + s[n - 2]/(n - 2); N[Table[s[k], {k, 1, z}]]; N[Table[k/(k - s[k]), {k, 2, z}], 20]; t = Table[r[n/(n - s[n]), (n + 1)/(n + 1 - s[n + 1])], {n, 2, z}]; fd = Denominator[t] (* Peter J. C. Moses, Jul 30 2013 *)

Formula

f(n) = n!/!n = A000142(n)/A000166(n). - Matthew House, Aug 14 2024

Extensions

Corrected and edited by Clark Kimberling, Jun 26 2015
Corrected and extended by Matthew House, Aug 14 2024

A227777 Least splitter of n-th and (n+1)st partial sums of 1/0! + 1/1! + ... + 1/n! + ... = e.

Original entry on oeis.org

1, 2, 3, 7, 39, 110, 252, 465, 1001, 9545, 27634, 136168, 589394, 398959, 5394991, 36568060, 130087267, 312129649, 5779594018, 5467464369, 69204258903, 186055048882, 403978495031, 8690849042711, 25668568633102, 246378923308185, 1163579759684330
Offset: 1

Views

Author

Clark Kimberling, Jul 30 2013

Keywords

Comments

Suppose x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Let s(n) = 1/0! + 1/1! + ... + 1/n!; since s(n) -> e, the corresponding least splitting rationals (see Example) also approach e.
Conjecture: a(n) <= n*sqrt(n!) for all n>0; see scatterplot under Links. - Jon E. Schoenfield, Jun 28 2015

Examples

			The first 19 splitting rationals are 2, 5/2, 8/3, 19/7, 106/39, 299/110, 685/252, 1264/465, 2721/1001, 25946/9545, 75117/27634, 370143/136168, 1602139/589394, 1084483/398959, 14665106/5394991, 99402293/36568060, 353613854/130087267, 848456353/312129649 & 15710565395/5779594018. Regarding the last one, |15710565395/5779594018 - e| < 10^(-19).
The numerators of these rationals are a proper subsequence of A006258 & A119014 and the denominators are a proper subsequence of A006259 & A119015. - _Robert G. Wilson v_, Jun 27 2015

Links

Crossrefs

Cf. A227631.

Programs

  • Mathematica
    z = 16; r[x_, y_] := Module[{a, b, x1 = Min[{x, y}], y1 = Max[{x, y}]}, If[x == y, x, b = NestWhile[#1 + 1 &, 1, ! (a = Ceiling[#1 x1 - 1]) < Ceiling[#1 y1] - 1 &]; (a + 1)/b]]; s[n_] := s[n] = Sum[1/(k - 1)!, {k, 1, n}]; N[Table[s[k], {k, 1, z}]]; t = Table[r[s[n], s[n + 1]], {n, 2, z}]; fd = Denominator[t] (* Peter J. C. Moses, Jul 20 2013 *)

Extensions

a(16)-a(17) from Manfred Scheucher, Jun 23 2015
a(18)-a(19) from Robert G. Wilson v, Jun 27 2015
a(20)-a(27) from Jon E. Schoenfield, Jun 27 2015

A227778 Least splitter of n-th and (n+1)st partial sums of 1/1 + 1/3 + ... + 1/(2n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 1, 13, 8, 6, 5, 4, 7, 3, 11, 8, 5, 7, 11, 2, 19, 11, 9, 7, 5, 8, 11, 20, 3, 13, 7, 11, 15, 4, 13, 9, 5, 16, 11, 6, 13, 7, 15, 8, 9, 10, 11, 13, 14, 17, 20, 24, 31, 43, 69, 1, 84, 49, 35, 27, 23, 19, 17, 15, 14, 12, 11, 21, 10, 9, 17, 8, 15
Offset: 1

Views

Author

Clark Kimberling, Jul 30 2013

Keywords

Comments

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. It appears that c/d is an integer (i.e., d = 1) for rationals in positions given by A082315; e.g. 1, 7, 56, ...

Examples

			The first 15 splitting rationals are 1/1, 3/2, 5/3, 7/4, 9/5, 17/9, 2/1, 27/13, 17/8, 13/6, 11/5, 9/4, 16/7, 7/3, 26/11.

Links

Crossrefs

Cf. A227631.

Programs

  • Mathematica
    z = 16; r[x_, y_] := Module[{a, b, x1 = Min[{x, y}], y1 = Max[{x, y}]}, If[x == y, x, b = NestWhile[#1 + 1 &, 1, ! (a = Ceiling[#1 x1 - 1]) < Ceiling[#1 y1] - 1 &]; (a + 1)/b]]; s[n_] := s[n] = Sum[1/(k - 1)!, {k, 1, n}]; N[Table[s[k], {k, 1, z}]]; t = Table[r[s[n], s[n + 1]], {n, 2, z}]; Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)

A227779 Least splitter of s(n) and s(n+1), where s(n) = sum{(k + 1/2)^(-1/2), k >= 1}.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 2, 3, 5, 1, 3, 2, 3, 5, 1, 4, 3, 2, 3, 6, 1, 4, 3, 2, 3, 5, 1, 5, 3, 2, 3, 4, 7, 1, 4, 3, 2, 3, 4, 6, 1, 5, 3, 5, 2, 3, 4, 7, 1, 5, 3, 5, 2, 3, 4, 6, 1, 6, 4, 3, 2, 5, 3, 5, 8, 1, 5, 4, 3, 2, 5, 3
Offset: 1

Views

Author

Clark Kimberling, Jul 30 2013

Keywords

Comments

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. It appears that d=1 (i.e., c/d is an integer) for rationals c/d in positions given by A024206.

Examples

			The first 15 splitting rationals are 1, 3/2, 2, 5/2, 3, 7/2, 11/3, 4, 9/2, 14/3, 5, 16/3, 11/2, 23/4, 6.

Links

Crossrefs

Cf. A227631.

Programs

  • Mathematica
    r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = Sum[(k + 1/2)^(-1/2), {k, 1, n}]; t = Table[r[s[n], s[n + 1]], {n, 1, 220}]; Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)

A227803 Least splitter of s(n) and s(n+1), where s(n) = (1 - 1/n)^n.

Original entry on oeis.org

1, 4, 10, 22, 3, 53, 35, 26, 23, 20, 37, 17, 48, 31, 45, 73, 14, 95, 67, 53, 39, 64, 25, 111, 61, 97, 36, 119, 83, 47, 105, 58, 69, 80, 91, 102, 124, 146, 179, 234, 322, 509, 11, 778, 448, 316, 250, 206, 173, 151, 140, 129, 118, 107, 203, 96, 181, 85, 159
Offset: 1

Views

Author

Clark Kimberling, Jul 31 2013

Keywords

Comments

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Since s(n) -> 1/e, the sequence of least splitting rationals also approaches 1/e .

Examples

			The first 15 splitting rationals are 0/1, 1/4, 3/10, 7/22, 1/3, 18/53, 12/35, 9/26, 8/23, 7/20, 13/37, 6/17, 17/48, 11/31, 16/45.

Links

Crossrefs

Cf. A227631, A225594.

Programs

  • Mathematica
    z = 100; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = (1 - 1/n)^n ; t = Table[r[s[n], s[n + 1]], {n, 1, z}]; Denominator[t] (* A227803, Peter J. C. Moses, Jul 15 2013 *)
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