A381747 a(n) is the number of solutions to tau(x) + tau(n-x) = tau(n) where 1 <= x <= floor(n/2).
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, 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, 1, 5, 0, 3, 0, 5, 8, 8, 0, 5, 0, 3, 0, 5, 0, 3, 1, 5, 0, 6
Offset: 1
Examples
a(10) = 3 because tau(x) + tau(10-x) = tau(10) has 3 solutions for 0 <= x <= 5: x = 1: tau(1) + tau(9) = 1 + 3 = 4 = tau(10); x = 3: tau(3) + tau(7) = 2 + 2 = 4 = tau(10); x = 5: tau(5) + tau(5) = 2 + 2 = 4 = tau(10).
Links
- Felix Huber, Table of n, a(n) for n = 1..10000
Programs
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Maple
with(NumberTheory): A381747:=proc(n) local a,x; a:=0; for x to n/2 do if tau(x)+tau(n-x)=tau(n) then a:=a+1 fi od; return a end proc; seq(A381747(n),n=1..88);
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PARI
a(n) = my(nd=numdiv(n)); sum(x=1, n\2, numdiv(x)+numdiv(n-x) == nd); \\ Michel Marcus, Apr 26 2025
Formula
a(A000430(n)) = 0 for n > 3.
Comments