cilleruelo {magic}

A class of multiplicative magic squares due to Cilleruelo and Luca

Description

Cilleruelo and Luca give a class of multiplicative magic squares whose behaviour is interesting.

Usage

cilleruelo(n, m)

Arguments

n, m

Arguments: usually integers

Details

\left( \begin{array}{cccc} (n+2)(m+0) & (n+3)(m+3) & (n+1)(m+2) & (n+0)(m+1)\\ (n+1)(m+1) & (n+0)(m+2) & (n+2)(m+3) & (n+3)(m+0)\\ (n+0)(m+3) & (n+1)(m+0) & (n+3)(m+1) & (n+2)(m+2)\\ (n+3)(m+2) & (n+2)(m+1) & (n+0)(m+0) & (n+1)(m+3) \end{array} \right)

Value

Returns a 4\times 4 matrix.

Author(s)

Robin K. S. Hankin

References

Javier Cilleruelo and Florian Luca 2010, “On multiplicative magic squares”, The Electronic Journal of Combinatorics vol 17, number 8

Examples

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)