A077475 Greedy powers of (8/13): Sum_{n>=1} (8/13)^a(n) = 1.

1, 2, 11, 14, 25, 28, 30, 37, 39, 41, 43, 46, 48, 51, 54, 57, 60, 64, 66, 71, 76, 78, 80, 82, 84, 90, 95, 101, 103, 106, 110, 113, 115, 117, 127, 133, 135, 140, 146, 152, 157, 160, 162, 165, 167, 170, 173, 179, 181, 185, 189, 196, 200, 203, 206, 209, 212, 215, 220

Offset: 1

Author

Paul D. Hanna, Nov 06 2002

Keywords

• easy
• nonn

Comments

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series Sum_{k=1..n} x^a(k) to exceed unity.

A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.8170308430..., where x=8/13 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002

By the time you reach Sum_{n=1..59} (8/13)^a(n), the difference between that sum and 1 is only 1.6*10^-47.

Examples

			a(3)=11 since (8/13) +(8/13)^2 +(8/13)^11 < 1 and (8/13)+(8/13)^2+(8/13)^10 >1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077472, A077473, A077474.

Programs

Maple

V:= Vector(100):
V[1]:= 1: T:= 1 - 8/13:
for n from 2 to 100 do
V[n]:= -floor(log[13/8](T));
T:= T - (8/13)^V[n];
od:
convert(V,list); # Robert Israel, Aug 11 2020

Mathematica

s = 0; a = {}; Do[ If[s + (8/13)^n < 1, s = s + (8/13)^n; a = Append[a, n]], {n, 1, 250}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[8/13], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) at x=8/13 and frac(y) = y - floor(y).

a(n) seems to be asymptotic to c*n with c around 3.7... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002.

Extended by Benoit Cloitre, Nov 06 2002