A077469 Greedy powers of (3/4): Sum_{n>=1} (3/4)^a(n) = 1.

1, 5, 16, 21, 29, 35, 39, 52, 57, 63, 68, 76, 82, 88, 93, 99, 106, 113, 118, 127, 134, 150, 155, 160, 167, 172, 182, 192, 197, 209, 215, 224, 229, 237, 242, 246, 260, 265, 272, 278, 289, 293, 310, 315, 320, 330, 337, 346, 353, 373, 379, 384, 390, 396, 405, 416

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) = 8.0547475948..., where x=3/4 and m=floor(log(1-x)/log(x))=4. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=9 since (3/4) +(3/4)^5 +(3/4)^16 < 1 and (3/4) +(3/4)^5 +(3/4)^15 > 1.
		

Crossrefs

Cf. A077468, A077470, A077471, A077472, A077473, A077474, A077475.

Programs

Mathematica

s = 0; a = {}; Do[ If[s + (3/4)^n < 1, s = s + (3/4)^n; a = Append[a, n]], {n, 1, 428}]; 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[3/4], 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=(3/4) and frac(y) = y - floor(y).

a(n) seems to be asymptotic to c*n with c around 8.0... - Benoit Cloitre

Extensions

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