A077468 Greedy powers of (2/3): sum_{n=1..inf} (2/3)^a(n) = 1.

1, 3, 9, 12, 15, 17, 27, 34, 39, 46, 49, 52, 54, 66, 70, 73, 81, 84, 90, 95, 102, 106, 110, 116, 119, 124, 132, 140, 143, 149, 153, 158, 161, 165, 171, 177, 180, 183, 186, 189, 194, 198, 209, 215, 221, 224, 226, 233, 235, 241, 244, 248, 251, 255, 259, 262, 272

Offset

1,2

Comments

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) = 4.9298413943..., where x=2/3 and m=floor(log(1-x)/log(x))=2. - Paul D. Hanna, Nov 16 2002

Links

Table of n, a(n) for n=1..57.

Michael Somos, Non-integer Radix Expansions and Modular Functions (1993)

Formula

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).

It appears that, for n>1, a(n)=A073536(n-1) - Benoit Cloitre, Jun 04 2004

Example

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).

Mathematica

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

Crossrefs

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

Sequence in context: A073105 A356101 A073536 * A089425 A255686 A138921

Adjacent sequences: A077465 A077466 A077467 * A077469 A077470 A077471

Keyword

easy,nonn

Author

Paul D. Hanna, Nov 06 2002

Extensions

Extended by John W. Layman, Robert G. Wilson v and Benoit Cloitre, Nov 07 2002

Status

approved