A181159 -id:A181159 - cp's OEIS frontend

A181160 Where records occur in A181159: a(n) occurs later than all positive integers k < a(n) in A181157.

1, 2, 3, 5, 7, 11, 13, 18, 23, 26, 31, 36, 41, 47, 54, 58, 63, 69, 72, 76, 84, 91, 95, 99, 104, 113, 118, 126, 133, 147, 158, 167, 170, 179, 192, 206, 229, 246, 258, 273, 283, 304, 316, 339, 369, 407, 432, 444, 470, 519, 533, 561, 608, 663

Offset: 1

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

a(34)-a(54) from Charlie Neder, Feb 07 2019

A306332 Records in A181159.

Original entry on oeis.org

1, 2, 6, 16, 36, 82, 166, 262, 382, 670, 766, 1066, 1492, 2050, 2470, 2710, 3370, 4192, 4432, 5050, 6274, 7354, 8050, 8590, 10270, 11256, 12250, 15190, 15670, 18910, 22666, 26026, 26110, 32146, 32230, 47806, 48250, 56026, 66652, 72766, 82630, 84706

Offset: 1

Charlie Neder, Feb 07 2019

nonn

A181160 gives the indices of these terms.

Cf. A181157, A181159, A181160.

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.

Original entry on oeis.org

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

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

nonn

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.

			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.

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

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

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 *)

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

cp's OEIS Frontend

A181160 Where records occur in A181159: a(n) occurs later than all positive integers k < a(n) in A181157.

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A306332 Records in A181159.

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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.

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