A270749 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A270749

(r,1)-greedy sequence, where r(k) = k/log(k+1).

2, 7, 117, 28231, 934841727, 1391154929853413822, 3358221400639080017571595039208647108, 84149630763494298099512446622134485046922136023978562834130778814722933257

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers. Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x. The sequence (a(n)) is the (r,x)-greedy sequence. We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x. See A270744 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..8.

FORMULA

a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).

r(1)/a(1) + ... + r(n)/a(n) + ... = 1

EXAMPLE

a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 2;

a(2) = ceiling(r(2)/(1 - r(1)/1) = 7;

a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2) = 117.

The first 3 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are

0.721..., 0.981..., 0.99991...

MATHEMATICA