

A225852


The number of people required for there to be at least a 50% chance that at least 3 share a birthday in a year with n days.


4



3, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 29, 30, 30
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OFFSET

1,1


COMMENTS

a(365) = 88.
For n <= 1000, a(n) = 1.436 + 1.812*n^0.654  0.817/n^3 provides an estimate accurate to 0.6 units.


LINKS

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


EXAMPLE

The probability that out of 87 people 3 share a birthday in a year with 365 days is 0.4994549. The corresponding probability for 88 people is 0.5110651. Therefore a(365)=88.


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, 3, y[i1]); while(BDaySharedByAtLeast(i, j, 3)<.5) j=j+1; y[i]=j}; y


CROSSREFS

Cf. A014088 (n people on 365 days), A033810 (2 people on n days), A225871 (4 people on n days).
Cf. A088141, A182008, A182009, A182010.
Sequence in context: A121857 A121854 A196119 * A198458 A134483 A121151
Adjacent sequences: A225849 A225850 A225851 * A225853 A225854 A225855


KEYWORD

nonn


AUTHOR

Kevin L. Schwartz and Christian N. K. Anderson, May 17 2013


STATUS

approved



