A382074 a(n) is the number of solutions to phi(x) + phi(n-x) = phi(n) where 1 <= x <= floor(n/2).

0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 3, 2, 2, 0, 2, 2, 2, 2, 4, 0, 0, 0, 1, 3, 1, 1, 2, 0, 3, 1, 4, 0, 1, 0, 5, 3, 2, 0, 2, 0, 2, 3, 5, 0, 2, 1, 5, 2, 1, 0, 1, 0, 2, 2, 1, 2, 2, 0, 5, 2, 2, 0, 3, 0, 2, 4, 5, 1, 3, 0, 4, 0, 1, 0, 2, 2, 2, 4, 5

Offset: 1

Author

Felix Huber, Mar 22 2025

Keywords

• nonn

Comments

If p is a prime and p != 3, then a(p) = 0. Proof: For p = 2, phi(1) + phi(1) = 2 > phi(2) = 1. For p > 3, phi(x) + phi(p-x) <= x - 1 + p - x - 1 = p - 2 < p - 1 = phi(p).

If a(2*i) = 0, then i is a positive odd number. Proof: If i is a positive even number, then 2*i = 2^k*(2*m-1) with integers k, m where k > 1 and m > 0. Since phi(2^k*(2*m-1)) = phi(2^k)*phi(2*m-1) = 2^(k-1)*phi(2*m-1) = 2*2^(k-2)*phi(2*m-1) = 2*phi(2^(k-1)*(2*m-1)), it follows that x = 2^(k-1)*(2*m-1) is a solution to phi(x) + phi(2^k*(2*m-1)-x) = phi(2^k*(2*m-1)).

a(2*i) = 0 is not true for every positive odd i. For example, a(2*3) = 1. It is conjectured that a(2*A065381(n)) = 0 for n > 1. However, there are positive odd numbers i not in A065381 and for which a(2*i) = 0. For example, a(2*529) = a(2*1155) = 0.

Examples

			a(20) = 3 because phi(x) + phi(20-x) = phi(20) has 3 solutions for 0 <= x <= 10:
  x = 6: phi(6) + phi(14) = 2 + 6 = 8 = phi(20).
  x = 8: phi(8) + phi(12) = 4 + 4 = 8 = phi(20).
  x = 10: phi(10) + phi(10) = 4 + 4 = 8 = phi(20).
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A065381, A211225, A381747.

Programs

Maple

with(NumberTheory):
A382074:=proc(n)
    local a,x;
    a:=0;
    for x to n/2 do
        if phi(x)+phi(n-x)=phi(n) then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A382074(n),n=1..88);

PARI

a(n) = my(e=eulerphi(n)); sum(x=1, n\2, eulerphi(x) + eulerphi(n-x) == e); \\ Michel Marcus, Mar 22 2025

Formula

a(p) = 0 for primes p != 3.

a(2^k*(2*m-1)) > 0 for integers k, m where k > 1 and m > 0.

Conjecture: a(2*A065381(n)) = 0 for n > 1.