A077471 Greedy powers of (4/7): sum_{n=1..inf} (4/7)^a(n) = 1.

1, 2, 5, 6, 10, 11, 14, 18, 19, 23, 27, 29, 30, 35, 36, 39, 55, 56, 60, 62, 64, 73, 75, 78, 79, 83, 84, 87, 95, 99, 104, 111, 113, 121, 122, 126, 133, 134, 141, 143, 147, 151, 152, 161, 162, 165, 169, 171, 173, 175, 176, 179, 182, 183, 186, 189, 197, 202, 205, 207

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) = 3.0486255758..., where x=4/7 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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=(4/7) and frac(y) = y - floor(y).

a(n) seems to be asymptotic to c*n with c around 3.3... - Benoit Cloitre

Example

a(3)=5 since (4/7) +(4/7)^2 +(4/7)^5 < 1 and (4/7) +(4/7)^2 +(4/7)^4 > 1.

Maple

s:= 0: count:= 0:
R:= NULL;
for n from 1 while count < 100 do
  t:= (4/7)^n;
  if s+t < 1 then count:= count+1; R:= R, n; s:= s+t fi
od:
R; # Robert Israel, Jun 01 2018

Mathematica

s = 0; a = {}; Do[ If[s + (4/7)^n < 1, s = s + (4/7)^n; a = Append[a, n]], {n, 1, 208}]; 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[4/7], 20]

Crossrefs

Cf. A077468, A077469, A077470, A077472, A077473, A077474, A077475.

Sequence in context: A187840 A187904 A179543 * A273324 A238096 A064572

Adjacent sequences: A077468 A077469 A077470 * A077472 A077473 A077474

Keyword

easy,nonn

Author

Paul D. Hanna, Nov 06 2002

Extensions

Extended by Benoit Cloitre, Nov 06 2002

Edited and extended by Robert G. Wilson v, Nov 08 2002

Status

approved