%I #12 Mar 16 2015 22:27:52

%S 1,4,8,18,21,28,31,36,41,44,55,58,71,76,79,84,88,108,125,135,141,148,

%T 155,158,164,175,180,185,195,198,218,225,230,237,242,246,250,254,259,

%U 263,268,276,281,300,305,310,317,321,326,329,334,340,343,351,359,364

%N Greedy powers of (7/10): sum_{n=1..inf} (7/10)^a(n) = 1.

%C 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.

%C A heuristic argument suggests that the limit of a(n)/n is m - sum_{n=m..inf} log(1 + x^n)/log(x) = 5.9293123466..., where x=7/10 and m=floor(log(1-x)/log(x))=3. - _Paul D. Hanna_, Nov 16 2002

%F 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=(7/10) and frac(y) = y - floor(y).

%F a(n) seems to be asymptotic to c*n with c around 6... - _Benoit Cloitre_

%e a(3)=8 since (7/10) +(7/10)^3 +(7/10)^8 < 1 and (7/10) +(7/10)^3 +(7/10)^7 > 1.

%t s = 0; a = {}; Do[ If[s + (7/10)^n < 1, s = s + (7/10)^n; a = Append[a, n]], {n, 1, 368}]; a

%t 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[7/10], 20]

%Y Cf. A077468, A077469, A077470, A077471, A077472, A077473, A077475.

%K easy,nonn

%O 1,2

%A _Paul D. Hanna_, Nov 06 2002

%E Edited and extended by _Robert G. Wilson v_, Nov 08 2002; also extended by _Benoit Cloitre_, Nov 06 2002