Python 2, 44 bytes

f=lambda n,x=1:+(x>n)or(n%x<1)-f(n,x+2)/4<<2

This is almost the same as xsot's solution (which is based on Peter Taylor's solution), but saves 8 bytes by simplifying the way signs are handled.

Note that for a full program, we can save 2 bytes in the function without incurring a cost outside the function:

f=lambda n,x=1:x>n or(n%x<1)-f(n,x+2)/4<<2
print+f(input())

Two additional bytes for a full program this way:

n=input()
f=lambda x:x>n or(n%x<1)-f(x+2)/4<<2
print+f(1)

For n > 0 there is a very legible 40-byte solution:

f=lambda n,x=1:n/x and(n%x<1)*4-f(n,x+2)

Python 2, 44 bytes

f=lambda n,x=1:+(x>n)or(n%x<1)-f(n,x+2)/4<<2

This is almost the same as xsot's solution (which is based on Peter Taylor's solution), but saves 8 bytes by simplifying the way signs are handled.

Note that for a full program, we can save 2 bytes in the function without incurring a cost outside the function:

f=lambda n,x=1:x>n or(n%x<1)-f(n,x+2)/4<<2
print+f(input())

Two additional bytes for a full program this way:

n=input()
f=lambda x:x>n or(n%x<1)-f(x+2)/4<<2
print+f(1)

For n > 0 there is a very legible 40-byte solution:

f=lambda n,x=1:n/x and(n%x<1)*4-f(n,x+2)

Python 2, 44 bytes

f=lambda n,x=1:+(x>n)or(n%x<1)-f(n,x+2)/4<<2

This is almost the same as xsot's solution (which is based on Peter Taylor's solution), but saves 8 bytes by simplifying the way signs are handled.

Note that for a full program, we can save 2 bytes in the function without incurring a cost outside the function:

f=lambda n,x=1:x>n or(n%x<1)-f(n,x+2)/4<<2
print+f(input())

For n > 0 there is a very legible 40-byte solution:

f=lambda n,x=1:n/x and(n%x<1)*4-f(n,x+2)

Python 2, 44 bytes

f=lambda n,x=1:+(x>n)or(n%x<1)-f(n,x+2)/4<<2

This is almost the same as xsot's solution (which is based on Peter Taylor's solution), but saves 8 bytes by simplifying the way signs are handled.

Note that for a full program, we can save 2 bytes in the function without incurring a cost outside the function:

f=lambda n,x=1:x>n or(n%x<1)-f(n,x+2)/4<<2
print+f(input())

Two additional bytes for a full program this way:

n=input()
f=lambda x:x>n or(n%x<1)-f(x+2)/4<<2
print+f(1)

For n > 0 there is a very legible 40-byte solution:

f=lambda n,x=1:n/x and(n%x<1)*4-f(n,x+2)