A077474 Greedy powers of (7/10): sum_{n=1..inf} (7/10)^a(n) = 1.

1, 4, 8, 18, 21, 28, 31, 36, 41, 44, 55, 58, 71, 76, 79, 84, 88, 108, 125, 135, 141, 148, 155, 158, 164, 175, 180, 185, 195, 198, 218, 225, 230, 237, 242, 246, 250, 254, 259, 263, 268, 276, 281, 300, 305, 310, 317, 321, 326, 329, 334, 340, 343, 351, 359, 364

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) = 5.9293123466..., where x=7/10 and m=floor(log(1-x)/log(x))=3. - 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=(7/10) and frac(y) = y - floor(y).

a(n) seems to be asymptotic to c*n with c around 6... - Benoit Cloitre

Example

a(3)=8 since (7/10) +(7/10)^3 +(7/10)^8 < 1 and (7/10) +(7/10)^3 +(7/10)^7 > 1.

Mathematica

s = 0; a = {}; Do[ If[s + (7/10)^n < 1, s = s + (7/10)^n; a = Append[a, n]], {n, 1, 368}]; 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[7/10], 20]

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077472, A077473, A077475.

Sequence in context: A312827 A308150 A070213 * A307817 A210433 A182099

Adjacent sequences: A077471 A077472 A077473 * A077475 A077476 A077477

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