log10（10の自然対数）の小数以下の数列で幾つかの数列パターンの、出現回数、出現箇所などを、探索してみた。
まずは、小数1000桁まで。
log10 = 2.302585092994045...2198849978
000: 1 @ 410
111: 0
222: 2 @ 226 295
333: 3 @ 56 519 806
444: 0
555: 1 @ 168
666: 1 @ 138
777: 1 @ 209
888: 1 @ 929
999: 0
123: 0
321: 1 @ 337
0, 1, ..., 9の出現回数
0: 103
1: 82
2: 106
3: 81
4: 106
5: 103
6: 107
7: 110
8: 106
9: 96
探索範囲を100万桁に拡げ、パターン長さを6に伸ばす。
log10 = 2.302585092994045...9665379150
000000: 1 @ 725886
111111: 1 @ 770997
222222: 1 @ 544115
333333: 4 @ 246249 504820 641772 911307
444444: 0
555555: 0
666666: 1 @ 674596
777777: 2 @ 691144 927426
888888: 1 @ 727419
999999: 1 @ 308195
123456: 2 @ 534364 712495
654321: 0
0, 1, ..., 9, 000, 111, ..., 999の出現回数
0: 99887
1: 100429
2: 99927
3: 100166
4: 100210
5: 99375
6: 99769
7: 99782
8: 100633
9: 99822
000: 905
111: 884
222: 897
333: 920
444: 917
555: 942
666: 903
777: 864
888: 918
999: 905
探索範囲を1億桁に拡げ、パターン長さを8に伸ばす。
log10 = 2.302585092994045...9017079147
00000000: 2 @ 68450558 90708701
11111111: 1 @ 91152051
22222222: 2 @ 6097484 99783570
33333333: 2 @ 58663526 70210995
44444444: 1 @ 66324140
55555555: 2 @ 22122117 85061011
66666666: 2 @ 19760830 29125828 55646198 84306619
77777777: 2 @ 24611354 65552532
88888888: 1 @ 64653957
99999999: 2 @ 41040896 88586210
12345678: 1 @ 75418751
87654321: 0
0, 1, ..., 9, 000, 111, ..., 999の出現回数
0: 9996731
1: 9999245
2: 10005133
3: 10001639
4: 9999948
5: 10000670
6: 10001182
7: 9999623
8: 9998524
9: 9997305
000: 90947
111: 90696
222: 91206
333: 91487
444: 91140
555: 91281
666: 90765
777: 90538
888: 90822
999: 90687