A181157 a(n) is the greatest integer such that the occurrence rate of integers k and above in {a(1), ..., a(n)} <= 1/k for all positive integers k.

1, 2, 1, 4, 1, 3, 1, 2, 1, 10, 1, 6, 1, 2, 1, 5, 1, 3, 1, 2, 1, 22, 1, 4, 1, 2, 1, 9, 1, 3, 1, 2, 1, 8, 1, 7, 1, 2, 1, 5, 1, 3, 1, 2, 1, 46, 1, 4, 1, 2, 1, 17, 1, 3, 1, 2, 1, 14, 1, 6, 1, 2, 1, 4, 1, 3, 1, 2, 1, 35, 1, 12, 1, 2, 1, 5, 1, 3, 1, 2, 1, 11, 1, 4, 1, 2, 1, 8, 1, 3

Offset

1,2

Comments

a(n) = 1 when n mod 2 = 1.

a(n) = 2 when n mod 6 = 2.

a(n) = 3 when n mod 12 = 6.

a(n) = 4 when n mod 60 = 4, 24 or 48.

a(n) = 5 when n mod 60 = 16 or 40.

Otherwise a(n) >= 6.

When we pick a term from this sequence at random, the expectation diverges to infinity.

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Example

Let denote the occurrence rate of integers k and above in {a(1), ..., a(n)} as r(n,k). For example, r(5,2) = 2/5 since {a(1), ..., a(5)} = {1, 2, 1, 4, 1}: integers 2 and above occur twice in them.
a(6) can be 3 or above since r(6,1) = 6/6 <= 1/1, r(6,2) = 3/6 <= 1/2, and r(6,3) = 2/6 <= 1/3. But if a(6) >= 4, then r(6,4) = 2/6 > 1/4. Thus a(6) cannot be greater than 3, therefore a(6) = 3.

Mathematica

mx = 60; acc = ConstantArray[0, mx + 1]; a = {}; Do[AppendTo[a, k = Min[Select[Range[mx], n/# - acc[[#]] < 1 &]] - 1]; acc[[Range[k]]]++, {n, mx}]; a (* Ivan Neretin, May 20 2015 *)

Crossrefs

Cf. A181158 (records), A181159 (first occurrence of n).

Sequence in context: A243792 A124331 A242885 * A095248 A122458 A329644

Adjacent sequences: A181154 A181155 A181156 * A181158 A181159 A181160

Keyword

nonn

Author

Keisuke Sato (st(AT)r3z.org), Oct 07 2010

Extensions

a(61)-a(90) added from b-file by Charlie Neder, Feb 08 2019

Status

approved