A014088 Minimal number of people to give a 50% probability of having at least n coincident birthdays in one year.

1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 1385, 1596, 1813, 2035, 2263, 2494, 2730, 2970, 3213, 3459, 3707, 3959, 4213, 4470, 4728, 4989, 5252, 5516, 5783, 6051, 6320, 6592, 6864, 7138, 7413, 7690, 7968, 8247, 8527, 8808, 9090, 9373, 9657, 9942, 10228

Offset: 1

Author

Steven Finch

Keywords

• nonn

Links

• Stig Blücher Brink, Table of n, a(n) for n = 1..10000 (terms 1..61 from Hiroaki Yamanouchi, terms 62..250 from Rob Cook)
• Patrice Le Conte, Coincident Birthdays.
• P. Diaconis and F. Mosteller, Methods of studying coincidences, J. Amer. Statist. Assoc. 84 (1989), pp. 853-861.
• Bruce Levin, Exact Solutions of the Generalized Birthday Problem.
• B. Martin, Coincidence:Remarkable or Random, Skeptical Inquirer Volume 22.5, September / October 1998.
• I. Peterson, Mathtrek, Birthday Surprises [Archived version from Jun 28 2013]
• Maksym Petkus, Efficient (Non-)Membership Tree from Multicollision-Resistance with Applications to Zero-Knowledge Proofs, Cryptology ePrint Archive (2024). See pp. 6, 34.
• Eric Weisstein's World of Mathematics, Birthday Problem.

Crossrefs

Cf. A033810 (2 people on n days), A225852 (3 people on n days), A225871 (4 people on n days).

Cf. A088141, A182008, A182009, A182010.

Programs

Mathematica

q[1][n_, d_] := q[1][n, d] = d!/((d-n)!*d^n) // N; q[k_][n_, d_] := q[k][n, d] = Sum[ n!*d!/(d^(i* k)*i!*(k!)^i*(n-i*k)!*(d-i)!)*Sum[ q[j][n-i*k, d-i]*(d-i)^(n-i* k)/d^(n-i*k), {j, 1, k-1}], {i, 1, Floor[n/k]}] // N; p[k_][n_, d_] := 1 - Sum[q[i][n, d], {i, 1, k-1}]; a[1] = 1; a[k_] := a[k] = For[n = a[k-1], True, n++, If[p[k][n, 365] >= 1/2, Return[n]]]; Table[ Print["a(", k, ") = ", a[k]]; a[k], {k, 1, 15}] (* Jean-François Alcover, Jun 12 2013, after Eric W. Weisstein *)

Extensions

Broken links corrected by Steven Finch, Jan 27 2009

a(16)-a(45) from Hiroaki Yamanouchi, Mar 19 2015