A080057 Greedy powers of (e^-Gamma): sum_{n=1..inf} (e^-Gamma)^a(n) = 1, where e^-Gamma = e^(-.57721566490153286...) = .561459483566885169...

1, 2, 4, 7, 9, 13, 15, 17, 20, 21, 23, 27, 29, 34, 35, 38, 40, 42, 43, 46, 48, 49, 51, 54, 57, 58, 61, 64, 65, 68, 73, 74, 80, 83, 85, 87, 89, 98, 100, 101, 104, 105, 107, 110, 113, 116, 117, 120, 122, 123, 126, 128, 132, 136, 139, 142, 149, 152, 156, 157, 160, 161, 163

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) = 2.909795625992782..., where x=(e^-Gamma) and m=floor(log(1-x)/log(x))=1.

Links

Table of n, a(n) for n=1..63.

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=(e^-Gamma) and frac(y) = y - floor(y). See A077468 for mathematica program by Robert G. Wilson v.

Example

a(3)=4 since (e^-Gamma) +(e^-Gamma)^2 +(e^-Gamma)^4 < 1 and (e^-Gamma) +(e^-Gamma)^2 +(e^-Gamma)^k > 1 for 2<k<4.

Crossrefs

Cf. A077468, A076802, A080056, A080058.

Sequence in context: A120878 A137996 A360627 * A087158 A195669 A277733

Adjacent sequences: A080054 A080055 A080056 * A080058 A080059 A080060

Keyword

easy,nonn

Author

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

Status

approved