Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #10 Oct 31 2013 12:23:01
%S 1,2,8,12,14,16,25,39,42,44,46,49,51,53,59,70,73,78,81,83,85,86,101,
%T 103,105,116,118,119,126,130,135,137,139,142,144,147,148,158,161,163,
%U 170,171,178,181,186,188,190,192,194,195,204,207,209,212,216,219,224,229
%N Greedy powers of (1/zeta(2)): sum_{n=1..inf} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...
%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..inf} log(1 + x^n)/log(x) = 3.66565771136..., where x=(1/zeta(2)) and m=floor(log(1-x)/log(x))=1.
%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=(1/zeta(2)) and frac(y) = y - floor(y). See A077468 for mathematica program by _Robert G. Wilson v_.
%e a(3)=8 since (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^8 < 1 and (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^k > 1 for 2<k<8.
%Y Cf. A077468, A080057, A080059, A229099.
%K nonn
%O 1,2
%A _Benoit Cloitre_ and _Paul D. Hanna_, Jan 23 2003