1,2

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..inf} log(1 + x^n)/log(x) = 3.8170308430..., where x=8/13 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna (pauldhanna(AT)juno.com), 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.

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=(8/13) and frac(y) = y - floor(y).

a(n) seems to be asymptotic to c*n with c around 3.7... - Benoit Cloitre (benoit7848c(AT)orange.fr)

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

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]

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

Sequence in context: A130288 A031192 A034039 this_sequence A061845 A091211 A012019

Adjacent sequences: A077472 A077473 A077474 this_sequence A077476 A077477 A077478

easy,nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 06 2002

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 08 2002. Also extended by Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 06 2002