A270752 (r,1)-greedy sequence, where r(k) = 1/(k*e).

1, 1, 1, 1, 1, 1, 1, 1, 262, 167395, 42355398928, 2986137074379747535250, 16334453331070842795541380956715272941358931, 334377619479874433401339085661668551899899040409749812309411639875183486098285324762070

Offset

1,9

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..14.

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

example:
a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;
a(2) = ceiling(r(2)/(1 - r(1)/1) = 1;
a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2) = 1.
The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ...  are
0.367..., 0.551..., 0.674..., 0.766..., 0,839..., 0.901...

Mathematica

$MaxExtraPrecision = Infinity; z = 16;
r[k_] := N[1/(k*E), 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, A068985.

Sequence in context: A250767 A264972 A241877 * A224556 A052033 A105008

Adjacent sequences: A270749 A270750 A270751 * A270753 A270754 A270755

Keyword

nonn,easy

Author

Clark Kimberling, Apr 09 2016

Status

approved