%I #74 Aug 23 2024 12:19:40
%S 1,23,88,187,313,460,623,798,985,1181,1385,1596,1813,2035,2263,2494,
%T 2730,2970,3213,3459,3707,3959,4213,4470,4728,4989,5252,5516,5783,
%U 6051,6320,6592,6864,7138,7413,7690,7968,8247,8527,8808,9090,9373,9657,9942,10228
%N Minimal number of people to give a 50% probability of having at least n coincident birthdays in one year.
%H Stig Blücher Brink, <a href="/A014088/b014088.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..61 from Hiroaki Yamanouchi, terms 62..250 from Rob Cook)
%H Patrice Le Conte, <a href="/A225852/a225852.pdf">Coincident Birthdays</a>.
%H P. Diaconis and F. Mosteller, <a href="http://statweb.stanford.edu/~cgates/PERSI/papers/mosteller89.pdf">Methods of studying coincidences</a>, J. Amer. Statist. Assoc. 84 (1989), pp. 853-861.
%H Bruce Levin, <a href="/A014088/a014088.txt">Exact Solutions of the Generalized Birthday Problem</a>.
%H B. Martin, <a href="https://www.csicop.org/si/show/coincidences_remarkable_or_random">Coincidence:Remarkable or Random</a>, Skeptical Inquirer Volume 22.5, September / October 1998.
%H I. Peterson, Mathtrek, <a href="https://web.archive.org/web/20130628034833/http://www.maa.org/mathland/mathtrek_11_23_98.html">Birthday Surprises</a> [Archived version from Jun 28 2013]
%H Maksym Petkus, <a href="https://eprint.iacr.org/2024/1259">Efficient (Non-)Membership Tree from Multicollision-Resistance with Applications to Zero-Knowledge Proofs</a>, Cryptology ePrint Archive (2024). See pp. 6, 34.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BirthdayProblem.html">Birthday Problem</a>.
%t 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_ *)
%Y Cf. A033810 (2 people on n days), A225852 (3 people on n days), A225871 (4 people on n days).
%Y Cf. A088141, A182008, A182009, A182010.
%K nonn
%O 1,2
%A _Steven Finch_
%E Broken links corrected by _Steven Finch_, Jan 27 2009
%E a(16)-a(45) from _Hiroaki Yamanouchi_, Mar 19 2015