To give you a feeling how the distribution of the numbers is developing, I give you the function values for all powers of ten from 1 to 12.
W_{0}(10): 3
W_{1}(10): 6
W_{2}(10): 0
W_{3}(10): 0
W_{4}(10): 0
W_{0}(100): 17
W_{1}(100): 71
W_{2}(100): 11
W_{3}(100): 0
W_{4}(100): 0
W_{0}(10^{3}): 108
W_{1}(10^{3}): 686
W_{2}(10^{3}): 201
W_{3}(10^{3}): 4
W_{4}(10^{3}): 0
W_{0}(10^{4}): 755
W_{1}(10^{4}): 6598
W_{2}(10^{4}): 2592
W_{3}(10^{4}): 54
W_{4}(10^{4}): 0
W_{0}(10^{5}): 5936
W_{1}(10^{5}): 63449
W_{2}(10^{5}): 29916
W_{3}(10^{5}): 698
W_{4}(10^{5}): 0
W_{0}(10^{6}): 48474
W_{1}(10^{6}): 614400
W_{2}(10^{6}): 328988
W_{3}(10^{6}): 8137
W_{4}(10^{6}): 0
W_{0}(10^{7}): 406270
W_{1}(10^{7}): 5952657
W_{2}(10^{7}): 3550745
W_{3}(10^{7}): 90324
W_{4}(10^{7}): 3
W_{0}(10^{8}): 3532031
W_{1}(10^{8}): 58088295
W_{2}(10^{8}): 37432690
W_{3}(10^{8}): 946964
W_{4}(10^{8}): 19
W_{0}(10^{9}): 31295358
W_{1}(10^{9}): 568932663
W_{2}(10^{9}): 390065916
W_{3}(10^{9}): 9705879
W_{4}(10^{9}): 183
W_{0}(10^{10}): 279591668
W_{1}(10^{10}): 5588087493
W_{2}(10^{10}): 4034529147
W_{3}(10^{10}): 97790090
W_{4}(10^{10}): 1601
W_{0}(10^{11}): 2521429242
W_{1}(10^{11}): 54968844332
W_{2}(10^{11}): 41532029309
W_{3}(10^{11}): 977682518
W_{4}(10^{11}): 14598
W_{0}(10^{12}): 22996137423
W_{1}(10^{12}): 541664112990
W_{2}(10^{12}): 425608837164
W_{3}(10^{12}): 9730782305
W_{4}(10^{12}): 130117
Note: Always the sum is not 10^{n} but 10^{n}  1, because W (1) is not defined.
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When we calculated a total analysis up to 10^{10} for the first time in the early years, we thought that, contrary to our theoretical estimates, which say that the asymptotic density for the Zweiwertzahlen is 1 and the asymptotic density for the Dreiwertzahlen is 0, there was a calculation error, because after the total analysis by the computer up to 10^{10} the proportion of Dreiwertzahlen increases.
That is a contradiction. To solve the problem we decided to do the total analysis up to 10^{12}.
The relief and the cheering was very great when we read the intermediate results on the screen in the middle of the night during the calculation and the proportion of Dreiwertzahlen  as had long been expected  finally fell. If this hadn't been the case, then we could have thrown away all of our theoretical estimates.
The reason this takes so long is that the function ln ln n increases extremely slowly and it takes a very long time before the function ln ln n can assert itself against multiplicative constants.
We expect that around 10^{24} the Zweiwertzahlen will overtake the Einswertzahlen, i.e. W_{2}(n) > W_{1}(n) for all n > around 10^{24}.
However, the Einswertzahlen offer very strong resistance against the Zweiwertzahlen. According to our extrapolations, it takes an extremely long time before the Zweiwertzahlen goes to the asyptotic density 1 and the Einswertzahlen goes to the asyptotic density 0 approach.
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In the next posts I will tell you something about our algorithm and prove a very important theorem from the Minimum theory.
