A248562 Least k such that log(3/2) - sum{1/(h*3^h), h = 1..k} < 1/6^n.

1, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103

Offset

1,2

Comments

This sequence provides insight into the manner of convergence of sum{1/(h*3^h), h = 1..k} to log(3/2). Since a(n+1) - a(n) is in {1,2} for n >= 1, the sequences A248563 and A248564 partition the positive integers.

References

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

Links

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

Example

Let s(n) = log(3/2) - sum{1/(h*3^h), h = 1..n}.  Approximations follow:
n ... s(n) ........ 1/6^n
1 ... 0.0721318 ... 0.166667
2 ... 0.0165762 ... 0.0277777
3 ... 0.0042305 ... 0.0046296
4 ... 0.0011441 ... 0.0007716
5 ... 0.0003210 ... 0.0001286
a(4) = 5 because s(5) < 1/6^4 < s(4).

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 *)

Crossrefs

Cf. A016578 (log(3/2)), A248563, A248564, A248559, A248565.

Sequence in context: A258575 A052490 A332076 * A117672 A194383 A186326

Adjacent sequences: A248559 A248560 A248561 * A248563 A248564 A248565

Keyword

nonn,easy

Author

Clark Kimberling, Oct 09 2014

Status

approved