A080056 Greedy powers of (2/Pi): Sum_{n=1..inf} (2/Pi)^a(n) = 1.

1, 3, 5, 16, 22, 24, 28, 34, 37, 43, 45, 49, 51, 54, 57, 59, 65, 68, 70, 74, 80, 88, 94, 97, 100, 103, 108, 111, 113, 116, 122, 127, 129, 132, 137, 141, 143, 148, 151, 156, 161, 164, 166, 172, 174, 177, 184, 189, 202, 204, 208, 213, 216, 219, 225, 227, 238, 247

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..inf} log(1 + x^n)/log(x) = 4.2164448079..., where x=(2/Pi) and m=floor(log(1-x)/log(x))=2.

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

Examples

			a(3)=5 since (2/Pi) +(2/Pi)^3 +(2/Pi)^5 < 1 and (2/Pi) +(2/Pi)^3 +(2/Pi)^k > 1 for 3<k<5.
		

Crossrefs

Cf. A077468, A080055, A080057.

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/Pi) and frac(y) = y - floor(y).