A077468 Greedy powers of (2/3): Sum_{n>=1} (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

Author

Paul D. Hanna, Nov 06 2002

Keywords

• easy
• nonn

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} 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

Examples

			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).
		

Links

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

Crossrefs

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

Programs

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]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) 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

Extensions

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