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

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

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