login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A077469 Greedy powers of (3/4): sum_{n=1..inf} (3/4)^a(n) = 1. 7
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 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 22 05:29 EDT 2019. Contains 328315 sequences. (Running on oeis4.)