A247779 -id:A247779 - cp's OEIS frontend

A247778 Least k such that e - (1 + 1/k)^k < 1/n.

1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 89

Offset: 1

Clark Kimberling, Sep 23 2014

a(n+1) - a(n) is in {1,2} for n >= 1.

			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.

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A247779, A247780.

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*)

a(n) = {k = 1; while ((exp(1) - (1 + 1/k)^k) >= 1/n, k++); k;} \\ Michel Marcus, Sep 25 2014

A247785 Numbers k such that A247784(k+1) = A247784(k).

Original entry on oeis.org

2, 6, 10, 13, 17, 21, 25, 29, 32, 36, 40, 44, 47, 51, 55, 59, 63, 66, 70, 74, 78, 82, 85, 89, 93, 97, 100, 104, 108, 112, 116, 119, 123, 127, 131, 135, 138, 142, 146, 150, 153, 157, 161, 165, 169, 172, 176, 180, 184, 187, 191, 195, 199, 203, 206, 210, 214

Offset: 1

Clark Kimberling, Sep 24 2014

A247785 and A247786 are a complementary pair.

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

Cf. A247779, A247784, A247786.

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*)

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

Clark Kimberling, Sep 23 2014

Complement of A247779.

			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.

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

Cf. A247778, A247779.

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*)

cp's OEIS Frontend

A247778 Least k such that e - (1 + 1/k)^k < 1/n.

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A247785 Numbers k such that A247784(k+1) = A247784(k).

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

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