A080057 Greedy powers of exp(-gamma): Sum_{n>=1} exp(-gamma)^a(n) = 1, where exp(-gamma) = exp(-.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

Author

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

Keywords

• easy
• nonn
• changed

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) = 2.909795625992782..., where x=exp(-gamma) and m=floor(log(1-x)/log(x))=1.

See A077468 for Mathematica program by Robert G. Wilson v.

Examples

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

Crossrefs

Cf. A077468, A076802, A080056, A080058.

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=(exp(-Gamma)) and frac(y) = y - floor(y).