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A270749 (r,1)-greedy sequence, where r(k) = k/log(k+1). 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

$MaxExtraPrecision = Infinity; z = 16;

r[k_] := N[k/Log[k + 1], 1000]; f[x_, 0] = x;

n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

x = 1; Table[n[x, k], {k, 1, z}]

N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]

CROSSREFS

Cf.  A001620, A270744.

Sequence in context: A326964 A034902 A101429 * A206151 A070521 A292433

Adjacent sequences:  A270746 A270747 A270748 * A270750 A270751 A270752

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 09 2016

STATUS

approved

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Last modified March 5 14:38 EST 2021. Contains 341823 sequences. (Running on oeis4.)