A080058 Greedy powers of (1/zeta(2)): Sum_{n>=1} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...

1, 2, 8, 12, 14, 16, 25, 39, 42, 44, 46, 49, 51, 53, 59, 70, 73, 78, 81, 83, 85, 86, 101, 103, 105, 116, 118, 119, 126, 130, 135, 137, 139, 142, 144, 147, 148, 158, 161, 163, 170, 171, 178, 181, 186, 188, 190, 192, 194, 195, 204, 207, 209, 212, 216, 219, 224, 229

Offset: 1

Author

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

Keywords

• 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) = 3.66565771136..., where x=(1/zeta(2)) and m=floor(log(1-x)/log(x))=1.

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

Examples

			a(3)=8 since (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^8 < 1 and (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^k > 1 for 2<k<8.
		

Crossrefs

Cf. A077468, A080057, A080059, A229099.

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=(1/zeta(2)) and frac(y) = y - floor(y).