A336385 Number of k = x*y such that phi(k) = n*(phi(x) + phi(y)).

2, 5, 15, 14, 7, 25, 7, 36, 30, 26, 15, 68, 2, 21, 125, 78, 9, 94, 7, 90, 52, 33, 38, 208, 15, 11, 74, 69, 3, 227, 2, 166, 66, 33, 129, 276, 2, 25, 101, 228, 7, 115, 8, 76, 329, 47, 28, 482, 19, 65, 40, 50, 7, 248, 99, 234, 46, 34, 20, 572, 2, 5, 426, 356, 16

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

Table of n, a(n) for n=1..65.

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

Cf. A000010, A057635, A336384, A336710.

Sequence in context: A146112 A146111 A146110 * A146109 A146108 A032841

Adjacent sequences: A336382 A336383 A336384 * A336386 A336387 A336388

Keyword

nonn

Author

Jinyuan Wang, Aug 01 2020

Status

approved