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Comments from Max Alekseyev (maxale(AT)gmail.com), May 26 2007: (Start)
"It is easy to see that eulerphi(x*y) = x*eulerphi(y) as soon as x divides y.
"Since n!! = n * (n-2)!!, if n divides (n-2)!! then a(n) = eulerphi(n!!) = eulerphi(n * (n-2)!!) = n * eulerphi((n-2)!!) = n * a(n-2).
"The only cases when n does not divide (n-2)!! are:
"1) n is prime. In this case n is coprime to (n-2)!!, implying that eulerphi(n*(n-2)!!) = eulerphi(n)*eulerphi((n-2)!!) = (n-1)*a(n-2)
"2) n=2p where p is odd prime. Then eulerphi(2p*(n-2)!!) = eulerphi(p)*eulerphi(2*(n-2)!!) = (p-1)*2*eulerphi((n-2)!!) = (n-2)*a(n-2)
"In the other cases we have the following five cases:
"1) n=p*q, where p and q are distinct odd factors >1. Then (n-2)!! contains both p,q as factors and hence is divisible by n.
"2) n=p^2 where p is odd prime. Then (n-2)!! contains p and 2p as factor and hence is divisible by n.
"3) n=2*p*q, where p and q are distinct factors >1. Then (n-2)!! contains 2p and 2q as factors and hence is divisible by n.
"4) n=2*p^2 where p is odd prime. Then (n-2)!! contains 2p and 4p as factors and hence is divisible by n.
"5) n=2*2^2. Then (n-2)!! = 6!! = 6*4*2 is divisible by n." (End)
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