A077473 Greedy powers of (5/9): Sum_{n>=1} (5/9)^a(n) = 1.

1, 2, 4, 6, 8, 11, 13, 18, 21, 24, 27, 28, 30, 32, 35, 37, 40, 43, 45, 50, 51, 59, 62, 64, 73, 76, 79, 82, 83, 86, 87, 93, 96, 99, 100, 103, 106, 108, 110, 112, 113, 117, 118, 121, 123, 126, 127, 131, 137, 139, 140, 143, 145, 146, 148, 154, 155, 157, 163, 165, 166

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) = 2.8326013771..., where x=5/9 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=4 since (5/9) +(5/9)^2 +(5/9)^4 < 1 and (5/9) +(5/9)^2 +(5/9)^3 > 1.
		

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077472, A077474, A077475.

Programs

Mathematica

s = 0; a = {}; Do[ If[s + (5/9)^n < 1, s = s + (5/9)^n; a = Append[a, n]], {n, 1, 173}]; 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[5/9], 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) (n>0) at x=5/9 and frac(y) = y - floor(y).

a(n) seems to be asymptotic to c*n with c around 2.8... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002. Also extended by Benoit Cloitre, Nov 06 2002