A077470 Greedy powers of (3/5): Sum_{n>=1} (3/5)^a(n) = 1.

1, 2, 7, 9, 13, 15, 18, 20, 22, 27, 31, 37, 39, 40, 49, 55, 57, 66, 68, 70, 71, 77, 79, 81, 82, 87, 94, 98, 104, 106, 107, 114, 117, 120, 121, 129, 133, 136, 138, 141, 150, 151, 157, 158, 163, 166, 169, 173, 181, 184, 192, 198, 199, 205, 207, 209, 213, 218, 224

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) = 3.5151141759..., where x=3/5 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=7 since (3/5) +(3/5)^2 +(3/5)^7 < 1 and (3/5) +(3/5)^2 +(3/5)^6 > 1.
		

Crossrefs

Cf. A077468, A077469, A077471, A077472, A077473, A077474, A077475.

Programs

Mathematica

s = 0; a = {}; Do[ If[s + (3/5)^n < 1, s = s + (3/5)^n; a = Append[a, n]], {n, 1, 226}]; 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[3/5], 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=3/5 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. Also extended by Benoit Cloitre, Nov 06 2002