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A381747 a(n) is the number of solutions to tau(x) + tau(n-x) = tau(n) where 1 <= x <= floor(n/2).

%I A381747 #17 Apr 26 2025 08:27:25
%S A381747 0,1,0,1,0,1,0,1,0,3,0,1,0,2,1,2,0,2,0,1,1,3,0,1,0,4,0,3,0,3,0,3,1,4,
%T A381747 0,1,0,2,1,2,0,3,0,5,6,4,0,0,0,5,0,5,0,5,1,4,0,4,0,2,0,3,6,4,0,5,0,8,
%U A381747 1,5,0,3,0,5,8,8,0,5,0,3,0,5,0,3,1,5,0,6
%N A381747 a(n) is the number of solutions to tau(x) + tau(n-x) = tau(n) where 1 <= x <= floor(n/2).
%C A381747 Observation: For even positive multiples of 48, k <= 17000, a(k) = 0 only for k = 1*48, 2304 = 48*48 and 3600 = 75*48. The next numbers k which are multiples of 48 and for which a(k) = 0 are 46656, 63504, 233280, 513216, 793152, 2286144, 3111696.
%H A381747 Felix Huber, <a href="/A381747/b381747.txt">Table of n, a(n) for n = 1..10000</a>
%F A381747 a(A000430(n)) = 0 for n > 3.
%e A381747 a(10) = 3 because tau(x) + tau(10-x) = tau(10) has 3 solutions for 0 <= x <= 5:
%e A381747   x = 1: tau(1) + tau(9) = 1 + 3 = 4 = tau(10);
%e A381747   x = 3: tau(3) + tau(7) = 2 + 2 = 4 = tau(10);
%e A381747   x = 5: tau(5) + tau(5) = 2 + 2 = 4 = tau(10).
%p A381747 with(NumberTheory):
%p A381747 A381747:=proc(n)
%p A381747     local a,x;
%p A381747     a:=0;
%p A381747     for x to n/2 do
%p A381747         if tau(x)+tau(n-x)=tau(n) then
%p A381747             a:=a+1
%p A381747         fi
%p A381747     od;
%p A381747     return a
%p A381747 end proc;
%p A381747 seq(A381747(n),n=1..88);
%o A381747 (PARI) a(n) = my(nd=numdiv(n)); sum(x=1, n\2, numdiv(x)+numdiv(n-x) == nd); \\ _Michel Marcus_, Apr 26 2025
%Y A381747 Cf. A000005, A000430, A211225, A382074.
%K A381747 nonn,easy
%O A381747 1,10
%A A381747 _Felix Huber_, Mar 30 2025