

A225871


Number of people required for there to be a 50% probability that at least 4 share a birthday in a year with n days.


4



4, 6, 7, 9, 10, 11, 12, 13, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 46, 47, 48, 48, 49, 50, 50, 51, 51, 52, 53, 53, 54, 54, 55, 56
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OFFSET

1,1


COMMENTS

a(365)=187.
For n<1000, the formula a(n) = 2.79 + 2.456*n^0.732  1.825/n provides an estimate of a(n) accurate to 0.82.


LINKS

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000
Christian N. K. Anderson, Table of n, exact probabilities of a(n)1 and a(n) for n = 1..1000.
P. Le Conte, Coincident Birthdays


EXAMPLE

For a year with 365 days, a(365), the probability that out of 186 people 4 of them share a birthday is 0.495825. The corresponding probability for 187 people is 0.502685, and therefore a(365)=187.


PROG

(R) library(gmp); #prob of a maximum of exactly k coincident birthdays is
BigQ<function(nday, p, k) { #nday=days in a year; p=people
if(p<k  nday<1) return(0)
if(k==1) return(prod(1(1:p1)/nday))
tot=0;
for(i in 1:floor(p/k)) {
q=(1as.bigz(i)/nday)^(pk*i) * prod((pas.bigz(1:(k*i))+1)/nday) * prod((ndayas.bigz(1:i)+1)/((1:i)*factorialZ(k)))
tot=tot+as.numeric(q)*ifelse(k*i<p & k>1, sum(sapply(2:k1, function(j) BigQ(ndayi, pk*i, j))), 1)
}
tot
}
BDaySharedByAtLeast<function(nday, people, k) {
if(nday<1  people<k) return(0)
if(k==1) return(prod(1(1:people1)/nday))
prob=1; for(j in 2:k1) prob=probBigQ(nday, people, j); prob
}
y=rep(0, 100); for(i in 1:100) { j=ifelse(i==1, 4, y[i1]); while(BDaySharedByAtLeast(i, j, 4)<.5) j=j+1; y[i]=j}; y


CROSSREFS

Cf. A014088 (n people on 365 days), A033810 (2 people on n days), A225852 (3 on n days).
Cf. A088141, A182008, A182009, A182010.
Sequence in context: A024555 A269330 A213627 * A288383 A001690 A105447
Adjacent sequences: A225868 A225869 A225870 * A225872 A225873 A225874


KEYWORD

nonn


AUTHOR

Kevin L. Schwartz and Christian N. K. Anderson, May 18 2013


STATUS

approved



