A077469 Greedy powers of (3/4): sum_{n=1..inf} (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

1,2

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

Links

Table of n, a(n) for n=1..56.

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=(3/4) and frac(y) = y - floor(y).

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

Example

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

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]

Crossrefs

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

Sequence in context: A124865 A090781 A191264 * A022140 A041855 A030691

Adjacent sequences: A077466 A077467 A077468 * A077470 A077471 A077472

Keyword

easy,nonn

Author

Paul D. Hanna, Nov 06 2002

Extensions

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

Status

approved