A050255 A Diaconis-Mosteller approximation to the Birthday problem function.

1, 23, 88, 187, 313, 459, 622, 797, 983, 1179, 1382, 1592, 1809, 2031, 2257, 2489, 2724, 2963, 3205, 3450, 3698, 3949, 4203, 4459, 4717, 4977, 5239, 5503, 5768, 6036, 6305, 6575, 6847, 7121, 7395, 7671, 7948, 8227, 8506, 8787, 9068, 9351, 9634, 9919, 10204

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

Comment from Stig Blücher Brink, May 18 2023: (Start)

Maximum relative approximation error for a(1) to a(10000) is 0.27%.

Maximum absolute approximation error for a(1) to a(10000) is 2126.

(End)

Links

• _Stig Blücher Brink_, Table of n, a(n) for n = 1..10000
• P. Diaconis and F. Mosteller, Methods of studying coincidences, J. Amer. Statist. Assoc. 84 (1989), pp. 853-861.
• Eric Weisstein's World of Mathematics, Birthday Problem

Crossrefs

Cf. A014088, A050256.

Programs

Mathematica

a[n_]:=Ceiling[x /. N[Solve[x Exp[-x/(365 n)]==(365^(n-1) n! Log[2] (1-x/(365 (n+1))))^(1/n), x, Reals]]]; Flatten[Table[a[n], {n, 15}]] (* Iain Fox, Oct 26 2018 *)

Formula

a(n) is ceiling(x), where x is the real solution to x*exp(-x/(365*n)) = (log(2)*365^(n-1)*n!*(1 - x/(365*(n+1))))^(1/n). - Iain Fox, Oct 26 2018

Extensions

a(43)-a(45) from Alois P. Heinz, May 17 2023