A088141 a(n) = the largest k such that, if k samples are taken from a group of n items, with replacement, a duplication is unlikely (p<1/2).

1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 2

Author

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Nov 06 2003

Keywords

• nonn

Comments

Related to the birthday paradox. This is essentially the same as A033810.

Examples

			a(365)=22 because if 22 people are sampled, it is unlikely that two have the same birthday; but if 23 are sampled, it is likely.
		

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A033810, A072829.

Programs

Mathematica

lst = {}; s = 1; Do[Do[If[Product[(n - i)/n, {i, j}] <= 1/2, If[j > s, s = j]; AppendTo[lst, j]; Break[]], {j, s, s + 1}], {n, 2, 86}]; lst (* Arkadiusz Wesolowski, Apr 29 2012 *)

Python

from math import comb, factorial
def A088141(n):
    def p(m): return comb(n,m)*factorial(m)<<1
    kmin, kmax = 0, 1
    while p(kmax) > n**kmax: kmax<<=1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if p(kmid) <= n**kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmin # Chai Wah Wu, Jan 21 2025

Extensions

Edited by Don Reble, Nov 07 2005