A080055 Greedy powers of log(2): sum_{n>=1} (log(2))^a(n) = 1.

1, 4, 8, 11, 15, 20, 23, 30, 38, 43, 49, 54, 60, 65, 72, 78, 85, 90, 93, 100, 104, 108, 111, 115, 118, 122, 128, 132, 140, 144, 147, 152, 156, 159, 171, 174, 178, 181, 188, 191, 196, 203, 206, 210, 213, 232, 244, 248, 256, 260, 265, 269, 272, 276, 285, 289, 293

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} log(1 + x^n)/log(x) = 5.7114827587..., where x = log(2) and m = floor(log(1-x)/log(x))=3.

Links

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

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=log(2) and frac(y) = y - floor(y). See A077468 for Mathematica program by Robert G. Wilson v.

Example

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.

Crossrefs

Cf. A077468, A080056.

Sequence in context: A311050 A311051 A311052 * A311053 A311054 A311055

Adjacent sequences: A080052 A080053 A080054 * A080056 A080057 A080058

Keyword

easy,nonn

Author

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

Status

approved