OFFSET
1,1
COMMENTS
If phi(x*y) = n*(phi(x) + phi(y)) and phi(x) <= phi(y), then phi(x) <= 2*n and phi(y) <= n*phi(x).
a(n) >= 1 because (k, x, y) = (4*n^2, 2*n, 2*n) is a solution.
If gcd(n, 6) = 1, then a(n) >= 2 because (k, x, y) = (12*n^2, 3*n, 4*n) is also a solution. Note that a(n) = 2 when n = 1, 13, 31, 37, 61, 73, 97, 103, 149, 151, 157, 181, ...
Conjecture: a(n) > 2 if n is composite.
LINKS
Zhang Sibao and Xi Xiaozhong, Positive integer solutions on phi(ab) = k*(phi(a) + phi(b))
MAPLE
a(2) = 5 because k = 16, 24, 36, 40 and 60 satisfy the equation.
PROG
(PARI) f(n) = floor(n*exp(Euler)*log(log(n^2))+2.5*n/log(log(n^2)));
a(n) = {if(n==1, return(2)); my(t, v=List([])); for(x=1, f(2*n), if((t=eulerphi(x)) <= 2*n, for(y=1, f(t=n*t), if(eulerphi(x*y) == t+n*eulerphi(y), listput(v, x*y))))); #Set(v); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Jinyuan Wang, Aug 01 2020
STATUS
approved