A077471 Greedy powers of (4/7): Sum_{n>=1} (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

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

Examples

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

Links

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

Crossrefs

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

Programs

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]

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

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

Extensions

Extended by Benoit Cloitre, Nov 06 2002

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