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

Author

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

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)

Extensions

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