A class of multiplicative magic squares due to Cilleruelo and Luca
Cilleruelo and Luca give a class of multiplicative magic squares whose behaviour is interesting.
cilleruelo(n, m)
n,m |
Arguments: usually integers |
a 4x4 array
Returns a 4x4 matrix.
Robin K. S. Hankin
Javier Cilleruelo and Florian Luca 2010, “On multiplicative magic squares”, The Electronic Journal of Combinatorics vol 17, \#N8
is.magic(cilleruelo(5,6))
is.magic(cilleruelo(5,6),func=prod)
f <- function(n){
jj <-
sapply(
seq(from=5,len=n),
function(i){cilleruelo(i,i-4)}
)
xM <- apply(jj,2,max)
xm <- apply(jj,2,min)
cbind(xM-xm , 5^(5/12)*xm^0.5 , 6*xm^0.5)
}
matplot(f(200),type='l',log='xy',xlab='n',ylab='')
legend(x="topleft",legend=c("xM-xm","5^(5/12).xm^(1/2)","6xm^(1/2)"),
lty=1:3,col=1:3)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.