A247985 -id:A247985 - cp's OEIS frontend

A247986 Numbers k such that A247985(k+1) - A247985(k) = 2.

2, 5, 9, 13, 16, 20, 23, 27, 30, 34, 37, 41, 44, 48, 52, 55, 59, 62, 66, 69, 73, 76, 80, 84, 87, 91, 94, 98, 101, 105, 108, 112, 115, 119, 123, 126, 130, 133, 137, 140, 144, 147, 151, 155, 158, 162, 165, 169, 172, 176, 179, 183, 186, 190, 194, 197, 201, 204

Offset: 1

Clark Kimberling, Sep 29 2014

Complement of A247987.

Clark Kimberling, Table of n, a(n) for n = 1..400

Cf. A247985, A247987.

z = 100; p[k_] := p[k] = Product[(k^2 + h)/(k^2 - h), {h, 1, k}] (* Finch p. 14 *)
N[Table[p[n] - E, {n, 2, z}]]
f[n_] := f[n] = Select[1 + Range[z], p[#] - E < 1/n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] ;  (* A247985 *)
v = Differences[u];
Flatten[Position[v, 2]]; (* A247986 *)
Flatten[Position[v, 3]]; (* A247987 *)

A247987 Numbers k such that A247985(k+1) - A247985(k) = 3.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 17, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 40, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 88, 89, 90, 92

Offset: 1

Clark Kimberling, Sep 29 2014

Complement of A247986.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A247985, A247986.

z = 100; p[k_] := p[k] = Product[(k^2 + h)/(k^2 - h), {h, 1, k}] (* Finch p. 14 *)
N[Table[p[n] - E, {n, 2, z}]]
f[n_] := f[n] = Select[1 + Range[z], p[#] - E < 1/n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] ;  (* A247985 *)
v = Differences[u];
Flatten[Position[v, 2]]; (* A247986 *)
Flatten[Position[v, 3]]; (* A247987 *)

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

Clark Kimberling, Sep 29 2014

			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).

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

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

$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 *)

cp's OEIS Frontend

A247986 Numbers k such that A247985(k+1) - A247985(k) = 2.

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A247987 Numbers k such that A247985(k+1) - A247985(k) = 3.

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Author

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Mathematica

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

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Mathematica