A393269 - OEIS

%I #10 Feb 18 2026 08:03:17

%S 5,0,0,7,6,5,5,1,8,7,1,4,5,1,8,6,5,2,9,5,0,8,4,4,0,9,3,5,2,6,7,5,3,3,

%T 7,0,0,4,1,5,7,7,2,5,5,3,9,8,3,0,6,6,7,1,8,9,2,8,7,8,1,2,4,1,7,0,7,7,

%U 8,3,9,8,2,5,5,4,3,7,6,1,1,3,6,1,8,2,4,4,8,8,6,4,0,9,6,9,6,4,6,8,8,5,3,0,7

%N Decimal expansion of the probability that everyone in a randomly selected group of 3064 persons share their birthday with at least one other person, not considering leap years.

%C 3064 is the least number of persons for which the probability exceeds 1/2.

%C This constant is a rational number: its numerator and denominator both have 7849 digits: 1.00618594635389...*10^7848 / 2.00929558595969...*10^7848.

%C The sequence has a period of 2.498547877282...*10^5706.

%H Mario Cortina Borja, <a href="https://doi.org/10.1111/j.1740-9713.2013.00705.x">The strong birthday problem</a>, Significance, Vol. 10, No. 6 (2013), pp. 18-20; <a href="https://rss.onlinelibrary.wiley.com/doi/full/10.1111/j.1740-9713.2013.00705.x">alternative link</a>.

%H Anirban DasGupta, <a href="https://doi.org/10.1016/j.jspi.2003.11.015">The matching, birthday and the strong birthday problem: a contemporary review</a>, Journal of Statistical Planning and Inference, Vol. 130, No. 1-2 (2005), pp. 377-389; <a href="https://www.math.ucdavis.edu/~tracy/courses/math135A/UsefullCourseMaterial/birthday.pdf">alternative link</a>.

%H Chijul B. Tripathy, <a href="https://arxiv.org/abs/2510.26056">The Strong Birthday Problem Revisited</a>, arXiv:2510.26056 [math.CO], 2025.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BirthdayProblem.html">Birthday Problem</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Birthday_problem#Generalizations">Birthday problem: Generalizations</a>.

%F Equals p(3064, 365), where p(n, m) = (m!*n!/m^n) * Sum_{k=0..m} (-1)^k * (m-k)^(n-k)/(k!*(m-k)!*(n-k)!) is the probability for the general case of n persons, with m days. 3064 is the least value of n for which p(n, 365) exceeds 1/2 (DasGupta, 2005).

%F Equivalently, p(m, n) = Sum_{k=1..floor(n/2)} binomial(m, k) * k! * T(n, k) / m^n, where T(n, k) = A008299(n, k) are the associated Stirling numbers of second kind (Tripathy, 2025).

%e 0.500765518714518652950844093526753370041577255398306...

%t p[n_, m_] := (m!*n!/m^n) * Sum[(-1)^k *(m-k)^(n-k)/(k!*(m-k)!*(n-k)!), {k, 0, Min[n, m]}];

%t RealDigits[p[3064, 365], 10, 120][[1]]

%o (PARI) p(n, m) = (m!*n!/m^n) * sum(k = 0, min(n, m), (-1)^k *(m-k)^(n-k)/(k!*(m-k)!*(n-k)!));

%o list(len) = digits(floor(p(3064, 365)*10^len));

%Y Cf. A008299, A333507, A343015, A380129, A393268, A393270, A393271.

%K nonn,cons,easy

%O 0,1

%A _Amiram Eldar_, Feb 08 2026