A276498 Number of trials T of an event that occurs with probability 1-1/n that must be repeated consecutively so that the probability that the event occurs on all T trials (i.e., (1-1/n)^T) is minimally higher than 1/n.

1, 3, 5, 7, 10, 13, 16, 19, 22, 25, 29, 32, 36, 39, 43, 47, 51, 54, 58, 62, 66, 71, 75, 79, 83, 87, 92, 96, 100, 105, 109, 114, 118, 123, 127, 132, 136, 141, 146, 150, 155, 160, 165, 169, 174, 179, 184, 189, 194, 199, 203, 208, 213, 218, 223, 228, 233

Offset

2,2

Links

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000

Formula

a(n) = round(log(1/n)/log(1-1/n)).

a(n) = n log n - (log n)/2 + O(1). - Charles R Greathouse IV, Sep 06 2016

Example

a(20) = round(log(1/20)/log(1-1/20)) = round(log(0.05)/log(0.95)) = round(58.4...) = 58.
If a phenomenon P occurs with a 95% probability on any given trial, how many trials T must be performed so that the probability that P occurs on all T trials reduces to 5%? 0.95^T = 0.05 => T = log(0.05)/log(0.95) = 58.4... = 58 (rounded).

Mathematica

Table[Round[Log[1/n]/Log[1 - 1/n]], {n, 2, 58}] (* Michael De Vlieger, Sep 06 2016 *)

Prog

(PARI) a(n) = round (log(1/n)/log(1-1/n)); \\ Michel Marcus, Sep 05 2016
(PARI) a(n)=-log(n)\/log1p(-1/n) \\ Charles R Greathouse IV, Sep 05 2016

Crossrefs

Sequence in context: A330072 A213510 A130257 * A186148 A079511 A092757

Adjacent sequences: A276495 A276496 A276497 * A276499 A276500 A276501

Keyword

nonn

Author

Talha Ali, Sep 05 2016

Extensions

Name and Example edited by Jon E. Schoenfield, Jul 16 2017

Status

approved