A080055 - OEIS

%I #15 Mar 16 2015 01:38:09

%S 1,4,8,11,15,20,23,30,38,43,49,54,60,65,72,78,85,90,93,100,104,108,

%T 111,115,118,122,128,132,140,144,147,152,156,159,171,174,178,181,188,

%U 191,196,203,206,210,213,232,244,248,256,260,265,269,272,276,285,289,293

%N Greedy powers of log(2): sum_{n>=1} (log(2))^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. A heuristic argument suggests that the limit of a(n)/n is m - sum_{n>=m} log(1 + x^n)/log(x) = 5.7114827587..., where x = log(2) and m = floor(log(1-x)/log(x))=3.

%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=log(2) and frac(y) = y - floor(y). See A077468 for Mathematica program by _Robert G. Wilson v_.

%e a(3)=8 since (log(2)) + (log(2))^4 + (log(2))^8 < 1 and (log(2)) + (log(2))^4 + (log(2))^k > 1 for 4 < k < 8.

%Y Cf. A077468, A080056.

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_ and _Paul D. Hanna_, Jan 23 2003