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 1-6 of 6 results.

A247779 Numbers k such that A247778(k+1) - A247778(k) = 1.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 25, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 71, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 103
Offset: 1

Views

Author

Clark Kimberling, Sep 23 2014

Keywords

Comments

Complement of A247780.

Examples

			The values of e - (1 + 1/k)^k for k = 1..8 are approximately 0.718282, 0.468282, 0.347911, 0.276876, 0.229962, 0.196655, 0.171782, 0.152497, so that the first 6 terms of A247778 are 1,2,4,5,6,8, and the first three terms of A247779 are 1,3,4.

Links

Crossrefs

Cf. A247778, A247780.

Programs

  • Mathematica
    z = 600; f[n_] := f[n] = Select[Range[z], E - (1 + 1/#)^# < 1/n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]       (*A247778*)
d1 = Flatten[Position[Differences[u], 1]] (*A247779*)
d2 = Flatten[Position[Differences[u], 2]] (*A247780*)

A247780 Numbers k such that A247778(k+1) - A247778(k) = 2.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 19, 22, 24, 27, 30, 33, 35, 38, 41, 44, 47, 49, 52, 55, 58, 61, 63, 66, 69, 72, 74, 77, 80, 83, 86, 88, 91, 94, 97, 100, 102, 105, 108, 111, 113, 116, 119, 122, 125, 127, 130, 133, 136, 138, 141, 144, 147, 150, 152, 155, 158, 161, 164
Offset: 1

Views

Author

Clark Kimberling, Sep 23 2014

Keywords

Comments

Complement of A247779.

Examples

			The values of e - (1 + 1/k)^k for k = 1..8 are approximately 0.718282, 0.468282, 0.347911, 0.276876, 0.229962, 0.196655, 0.171782, 0.152497, so that the first 6 terms of A247778 are 1,2,4,5,6,8, and the first three terms of A247780 are 1,3,4.

Links

Crossrefs

Cf. A247778, A247779.

Programs

  • Mathematica
    z = 600; f[n_] := f[n] = Select[Range[z], E - (1 + 1/#)^# < 1/n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]       (*A247778*)
d1 = Flatten[Position[Differences[u], 1]] (*A247779*)
d2 = Flatten[Position[Differences[u], 2]] (*A247780*)

A247784 a(n) = floor(1/(e - (1 + 1/n)^n)).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 41, 41, 42, 43, 44, 44, 45, 46, 47, 47, 48, 49, 49, 50
Offset: 1

Views

Author

Clark Kimberling, Sep 24 2014

Keywords

Comments

a(n+1) - a(n) is in {0,1} for n >= 0.

Links

Crossrefs

Cf. A247778, A247785, A247786.

Programs

  • Magma
    [Floor(1/(Exp(1) - (1 + 1/n)^n)): n in [1..100]]; // G. C. Greubel, Sep 14 2018
  • Mathematica
    z = 200; t = Table[Floor[1/(E - (1 + 1/k)^k)], {k, 1, z}]   (*A247784*)
d = Differences[t]
Flatten[Position[d, 0]]  (*A247785*)
Flatten[Position[d, 1]]  (*A247786*)
  • PARI
    a(n) = floor(1/(exp(1) - (1 + 1/n)^n)); \\ Michel Marcus, Sep 26 2014

Formula

a(n) ~ 2*exp(-1) * n. - Vaclav Kotesovec, Oct 09 2014

A247786 Numbers k such that A247784(k+1) - A247784(k) = 1.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90
Offset: 1

Views

Author

Clark Kimberling, Sep 24 2014

Keywords

Comments

A247785 and A247786 are a complementary pair.

Links

Crossrefs

Cf. A247778, A247784, A247785.

Programs

  • Mathematica
    z = 200; t = Table[Floor[1/(E - (1 + 1/k)^k)], {k, 1, z}]  (*A247784*)
d = Differences[t]
Flatten[Position[d, 0]]  (*A247785*)
Flatten[Position[d, 1]]  (*A247786*)

A247781 Least k such that 1/e - (1 - 1/k)^k < 1/n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15
Offset: 1

Views

Author

Clark Kimberling, Sep 24 2014

Keywords

Comments

a(n+1) - a(n) is in {0,1} for n >= 1.

Examples

			The values of 1/e - (1 - 1/k)^k for n = 1..9 are approximately 0.367879, 0.117879, 0.0715831, 0.0514732, 0.0401994, 0.0329815, 0.0279628, 0.0242705, 0.02144, from which we see that the first 9 terms of A247781 are 1, 1, 2, 2, 2, 2, 2, 2, 3.

Links

Crossrefs

Cf. A247782, A247783, A247778.

Programs

  • Mathematica
    z = 400; f[n_] := f[n] = Select[Range[z], 1/E - (1 - 1/#)^# < 1/n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] (*A247781*)
d1 = Flatten[Position[Differences[u], 0]] (*A247782*)
d2 = Flatten[Position[Differences[u], 1]] (*A247783*)

A247988 Least number k such that e - k/(k!)^(1/k) < 1/n.

Original entry on oeis.org

4, 11, 19, 27, 36, 45, 54, 64, 74, 84, 94, 105, 115, 126, 136, 147, 158, 169, 180, 191, 203, 214, 225, 237, 248, 260, 272, 283, 295, 307, 319, 331, 343, 355, 367, 379, 391, 403, 416, 428, 440, 452, 465, 477, 490, 502, 515, 527, 540, 552, 565, 578, 590, 603
Offset: 1

Views

Author

Clark Kimberling, Sep 29 2014

Keywords

Examples

			Let w(n) = e - n/(n!)^(1/n).  Approximations are shown here:
n .... w(n)  ...... 1/n
1 .... 1.71828 .... 1
2 .... 1.30407 .... 0.5
3 .... 1.06732 .... 0.333333
4 .... 0.911078 ... 0.25
5 .... 0.799022 ... 0.2
10 ... 0.510157 ... 0.1
11 ... 0.477609 ... 0.090909
a(2) = 11 because w(11) < 1/2 < w(10).

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 14.

Crossrefs

Cf. A247778, A247908, A247911, A247914, A247985.

Programs

  • Mathematica
    $MaxExtraPrecision = Infinity;
z = 1000; p[k_] := p[k] = k/(k!)^(1/k) (* Finch p. 14 *)
N[Table[E - p[n], {n, 1, z}]];
f[n_] := f[n] = Select[Range[z], E - p[#] < 1/n &, 1];
u = Flatten[Table[f[n], {n, 1, z/10}]]  (* A247988 *)
Showing 1-6 of 6 results.

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