A077468 - OEIS

%I #17 Mar 12 2021 22:24:42

%S 1,3,9,12,15,17,27,34,39,46,49,52,54,66,70,73,81,84,90,95,102,106,110,

%T 116,119,124,132,140,143,149,153,158,161,165,171,177,180,183,186,189,

%U 194,198,209,215,221,224,226,233,235,241,244,248,251,255,259,262,272

%N Greedy powers of (2/3): sum_{n=1..inf} (2/3)^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) = 4.9298413943..., where x=2/3 and m=floor(log(1-x)/log(x))=2. - _Paul D. Hanna_, Nov 16 2002

%H Michael Somos, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/NEWS/exp-sqrt">Non-integer Radix Expansions and Modular Functions</a> (1993)

%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=(2/3) and frac(y) = y - floor(y).

%F It appears that, for n>1, a(n)=A073536(n-1) - _Benoit Cloitre_, Jun 04 2004

%e a(3)=9 since (2/3) +(2/3)^3 +(2/3)^9 < 1 and (2/3) +(2/3)^3 +(2/3)^8 > 1; since the power 8 makes the sum > 1, then 9 is the 3rd greedy power of (2/3).

%t s = 0; a = {}; Do[ If[s + (2/3)^n < 1, s = s + (2/3)^n; a = Append[a, n]], {n, 1, 278}]; 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[2/3], 20]

%Y Cf. A077469, A077470, A077471, A077472, A077473, A077474, A077475.

%K easy,nonn

%O 1,2

%A _Paul D. Hanna_, Nov 06 2002

%E Extended by _John W. Layman_, _Robert G. Wilson v_ and _Benoit Cloitre_, Nov 07 2002