A393269 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.

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, 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, 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

Offset

0,1

Comments

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

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

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

Links

Table of n, a(n) for n=0..104.

Mario Cortina Borja, The strong birthday problem, Significance, Vol. 10, No. 6 (2013), pp. 18-20; alternative link.

Anirban DasGupta, The matching, birthday and the strong birthday problem: a contemporary review, Journal of Statistical Planning and Inference, Vol. 130, No. 1-2 (2005), pp. 377-389; alternative link.

Chijul B. Tripathy, The Strong Birthday Problem Revisited, arXiv:2510.26056 [math.CO], 2025.

Eric Weisstein's World of Mathematics, Birthday Problem.

Wikipedia, Birthday problem: Generalizations.

Formula

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).

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).

Example

0.500765518714518652950844093526753370041577255398306...

Mathematica

p[n_, m_] := (m!*n!/m^n) * Sum[(-1)^k *(m-k)^(n-k)/(k!*(m-k)!*(n-k)!), {k, 0, Min[n, m]}];
RealDigits[p[3064, 365], 10, 120][[1]]

Prog

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

Crossrefs

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

Sequence in context: A388217 A261852 A254293 * A263496 A308224 A200630

Adjacent sequences: A393266 A393267 A393268 * A393270 A393271 A393272

Keyword

nonn,cons,easy

Author

Amiram Eldar, Feb 08 2026

Status

approved