A077472 Greedy powers of (5/8): Sum_{n>=1} (5/8)^a(n) = 1.

1, 3, 5, 8, 10, 13, 15, 23, 26, 30, 33, 36, 38, 46, 48, 51, 53, 57, 61, 64, 66, 69, 72, 76, 78, 84, 88, 93, 95, 104, 106, 110, 115, 117, 121, 126, 129, 131, 136, 138, 143, 148, 150, 152, 157, 160, 164, 169, 172, 175, 179, 181, 185, 187, 191, 196, 198, 201, 203

Offset: 1

Author

Paul D. Hanna, Nov 06 2002

Keywords

• easy
• nonn

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) = 3.9944918847..., where x=5/8 and m=floor(log(1-x)/log(x))=2. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=5 since (5/8) +(5/8)^3 +(5/8)^5 < 1 and (5/8) +(5/8)^3 +(5/8)^4 > 1.
		

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077473, A077474, A077475.

Programs

Mathematica

s = 0; a = {}; Do[ If[s + (5/8)^n < 1, s = s + (5/8)^n; a = Append[a, n]], {n, 1, 210}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[5/8], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) at x=(5/8) and frac(y) = y - floor(y).

a(n) seems to be asymptotic to c*n with c around 3.4... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002. Also extended by Benoit Cloitre, Nov 06 2002