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 1-4 of 4 results.

A248559 Least k such that log(2) - sum{1/(h*2^h), h = 1..k} < 1/3^n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 88, 89, 91, 92, 94, 95, 97, 98
Offset: 1

Views

Author

Clark Kimberling, Oct 09 2014

Keywords

Comments

This sequence provides insight into the manner of convergence of sum{1/(h*2^h), h = 1..k} to log 2. Since a(n+1) - a(n) is in {1,2} for n >= 1, the sequences A248560 and A248561 partition the positive integers.

Examples

			Let s(n) = log(2) - sum{1/(h*2^h), h = 1..n}.  Approximations follow:
n ... s(n) ........ 1/3^n
1 ... 0.193147 .... 0.33333
2 ... 0.0681472 ... 0.11111
3 ... 0.0264805 ... 0.037037
4 ... 0.0108555 ... 0.0123457
5 ... 0.0046066 ... 0.004115
6 ... 0.0020013 ... 0.00137174
a(5) = 6 because s(6) < 1/3^5 < s(5).

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 15.

Links

Crossrefs

Cf. A002162 (log(2)), A248560, A248561.

Programs

  • Mathematica
    z = 200; p[k_] := p[k] = Sum[1/(h*2^h), {h, 1, k}]
N[Table[Log[2] - p[n], {n, 1, z/5}]]
f[n_] := f[n] = Select[Range[z], Log[2] - p[#] < 1/3^n &, 1]
u = Flatten[Table[f[n], {n, 1, z}]]    (* A248559 *)
Flatten[Position[Differences[u], 1]]   (* A248560 *)
Flatten[Position[Differences[u], 2]]   (* A248561 *)

A248560 Numbers k such that A248559(k+1) = A248559(k) + 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 11, 13, 15, 17, 19, 21, 23, 25, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 50, 52, 54, 56, 59, 61, 63, 65, 68, 70, 72, 75, 77, 79, 82, 84, 86, 88, 91, 93, 95, 98, 100, 102, 105, 107, 109, 112, 114, 116, 119, 121, 123, 126, 128, 130
Offset: 1

Views

Author

Clark Kimberling, Oct 09 2014

Keywords

Examples

			(A248559(k+1) - A248559(k)) = (1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2,...), so that A248561 = (1, 2, 3, 5, 6, 8, 10, 11, 13, ...) and A248562 = (4, 7, 9, 12, 14, 16, 18, 20, ...).

Links

Crossrefs

Cf. A248559, A248561.

Programs

  • Mathematica
    z = 200; p[k_] := p[k] = Sum[1/(h*2^h), {h, 1, k}]
N[Table[Log[2] - p[n], {n, 1, z/5}]]
f[n_] := f[n] = Select[Range[z], Log[2] - p[#] < 1/3^n &, 1]
u = Flatten[Table[f[n], {n, 1, z}]]    (* A248559 *)
Flatten[Position[Differences[u], 1]]   (* A248560 *)
Flatten[Position[Differences[u], 2]]   (* A248561 *)

A248567 Numbers k such that A248565(k+1) = A248565(k) + 2.

Original entry on oeis.org

3, 6, 9, 11, 13, 16, 18, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 117, 119, 121, 123, 125, 127, 129
Offset: 1

Views

Author

Clark Kimberling, Oct 09 2014

Keywords

Examples

			(A248565(k+1) - A248565(k)) = (1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2,...), so that A248566 = (1, 2, 4, 5, 7, 8, 10, 12, 14, ..) and A248567 = (3, 6, 9, 11, 13, 16, 18, 20, ...).

Links

Crossrefs

Cf. A248565, A248566, A248561, A248564.

Programs

  • Mathematica
    z = 2500; p[k_] := p[k] = Sum[1/(h*4^h), {h, 1, k}];
N[Table[p[k], {k, 1, z/5}], 12];
N[Table[Log[4/3] - p[n], {n, 1, z/5}]];
f[n_] := f[n] = Select[Range[z], Log[4/3] - p[#] < 1/8^n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] ;   (* A248565 *)
Flatten[Position[Differences[u], 1]];   (* A248566 *)
Flatten[Position[Differences[u], 2]];   (* A248567 *)

A248564 Numbers k such that A248562(k+1) = A248562(k) + 2.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 16, 18, 20, 21, 23, 25, 26, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106, 108
Offset: 1

Views

Author

Clark Kimberling, Oct 09 2014

Keywords

Examples

			(A248562(k+1) - A248562(k)) = (1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2,...), so that A248563 = (1, 2, 4, 6, 8, 10, 12, 15, 17, ..) and A248564 = (3, 5, 7, 9, 11, 13, 14, 16, ...).

Links

Crossrefs

Cf. A248562, A248563, A248561, A248567.

Programs

  • Mathematica
    z = 300; p[k_] := p[k] = Sum[1/(h*3^h), {h, 1, k}];
N[Table[Log[3/2] - p[n], {n, 1, z/5}]]
f[n_] := f[n] = Select[Range[z], Log[3/2] - p[#] < 1/6^n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]    (* A248562 *)
Flatten[Position[Differences[u], 1]]   (* A248563 *)
Flatten[Position[Differences[u], 2]]   (* A248564 *)
Showing 1-4 of 4 results.

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