cp's OEIS Frontend

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

%I A382074 #13 Apr 26 2025 03:32:51
%S A382074 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,
%T A382074 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,
%U A382074 2,2,0,3,0,2,4,5,1,3,0,4,0,1,0,2,2,2,4,5
%N A382074 a(n) is the number of solutions to phi(x) + phi(n-x) = phi(n) where 1 <= x <= floor(n/2).
%C A382074 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).
%C A382074 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)).
%C A382074 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.
%H A382074 Felix Huber, <a href="/A382074/b382074.txt">Table of n, a(n) for n = 1..10000</a>
%F A382074 a(p) = 0 for primes p != 3.
%F A382074 a(2^k*(2*m-1)) > 0 for integers k, m where k > 1 and m > 0.
%F A382074 Conjecture: a(2*A065381(n)) = 0 for n > 1.
%e A382074 a(20) = 3 because phi(x) + phi(20-x) = phi(20) has 3 solutions for 0 <= x <= 10:
%e A382074   x = 6: phi(6) + phi(14) = 2 + 6 = 8 = phi(20).
%e A382074   x = 8: phi(8) + phi(12) = 4 + 4 = 8 = phi(20).
%e A382074   x = 10: phi(10) + phi(10) = 4 + 4 = 8 = phi(20).
%p A382074 with(NumberTheory):
%p A382074 A382074:=proc(n)
%p A382074     local a,x;
%p A382074     a:=0;
%p A382074     for x to n/2 do
%p A382074         if phi(x)+phi(n-x)=phi(n) then
%p A382074             a:=a+1
%p A382074         fi
%p A382074     od;
%p A382074     return a
%p A382074 end proc;
%p A382074 seq(A382074(n),n=1..88);
%o A382074 (PARI) a(n) = my(e=eulerphi(n)); sum(x=1, n\2, eulerphi(x) + eulerphi(n-x) == e); \\ _Michel Marcus_, Mar 22 2025
%Y A382074 Cf. A000010, A065381, A211225, A381747.
%K A382074 nonn
%O A382074 1,14
%A A382074 _Felix Huber_, Mar 22 2025