A248636 Least k such that 33/8- sum{(h^3)/3^h, h = 1..k} < 1/4^n.

7, 9, 10, 12, 13, 15, 17, 18, 19, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 90, 92, 93, 94

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• easy

Comments

This sequence provides insight into the manner of convergence of sum{(h^3)/3^h, h = 1..k} to 33/8. The difference sequence of A248636 entirely of 1s and 2s, so that A248637 and A248638 partition the positive integers.

Examples

			Let s(n) = 33/8- sum{(h^3)/3^h, h = 1..n}.  Approximations follow:
n ... s(n) ........ 1/4^n
1 ... 3.79167 ... 0.250000
2 ... 2.90278 ... 0.062500
3 ... 1.90278 ... 0.015625
4 ... 1.11265 ... 0.003906
5 ... 0.59825 ... 0.000976
6 ... 0.30195 ... 0.000244
7 ... 0.14511 ... 0.000061
8 ... 0.06798 ... 0.000015
9 ... 0.03004 ... 0.000004
a(2) = 9 because s(9) < 1/16 < s(8).
		

Links

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

Crossrefs

Cf. A248632, A248630.

Programs

Mathematica

z = 300; p[k_] := p[k] = Sum[(h^3/3^h), {h, 1, k}];
d = N[Table[33/8 - p[k], {k, 1, z/5}], 12]
f[n_] := f[n] = Select[Range[z], 33/8 - p[#] < 1/4^n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]  (* A248636 *)
d = Differences[u]
v = Flatten[Position[d, 1]] (* A248637 *)
w = Flatten[Position[d, 2]] (* A248638 *)