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Date: 2-1-2021
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Theodorus's constant has decimal expansion
(OEIS A002194). It was computed to decimal digits by E. Weisstein on Jul. 23, 2013.
The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874).
-constant primes occur at 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344) decimal digits.
The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 5, 1, 4, 3, 23, 6, 12, 2, 8, 18, ... (OEIS A229200).
Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 4, 91, 184, 5566, 86134, 35343, ... (OEIS A000000), which end at digits 23, 378, 7862, 77437, 1237533, 16362668, ... (OEIS A000000).
The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does, starting at positions 1104282392, 1879095207, 3037917993, ... (OEIS A000000) (E. Weisstein, Jul. 23, 2013).
It is not known if is normal (Beyer et al. 1969, 1970ab), but the following table giving the counts of digits in the first terms shows that the decimal digits are very uniformly distributed up to at least .
OEIS | 10 | 100 | |||||||||
0 | A000000 | 3 | 15 | 95 | 1035 | 10125 | 100234 | 1000172 | 9995281 | 99976638 | 1000006042 |
1 | A000000 | 0 | 7 | 97 | 996 | 10019 | 99587 | 1001548 | 10001670 | 99988551 | 999978902 |
2 | A000000 | 1 | 8 | 100 | 994 | 9829 | 99812 | 1000263 | 10001751 | 99991487 | 999982296 |
3 | A000000 | 1 | 9 | 97 | 945 | 9898 | 99818 | 998943 | 10000247 | 100004464 | 999998469 |
4 | A000000 | 0 | 7 | 84 | 971 | 10077 | 99897 | 998647 | 10001384 | 100023203 | 1000009144 |
5 | A000000 | 2 | 13 | 93 | 1009 | 10037 | 100260 | 999993 | 9995879 | 99996674 | 999982506 |
6 | A000000 | 0 | 10 | 103 | 1027 | 10052 | 100558 | 999976 | 9999931 | 100020148 | 1000025094 |
7 | A000000 | 2 | 11 | 98 | 991 | 9921 | 99921 | 1000059 | 10002655 | 99987934 | 999997927 |
8 | A000000 | 1 | 14 | 125 | 1002 | 9996 | 100055 | 1000650 | 10001042 | 100017107 | 1000013674 |
9 | A000000 | 0 | 6 | 108 | 1030 | 10046 | 99858 | 999749 | 10000160 | 99993794 | 1000005946 |
REFERENCES:
Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent." Math. Comput. 23, 679, 1969.
Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Statistical Study of Digits of Some Square Roots of Integers in Various Bases." Math. Comput. 24, 455-473, 1970a.
Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases." Math. Comput. 24, 745-747, 1970b.
Sloane, N. J. A. Sequences A002194/M4326, A119344, A224874, A229200 in "The On-Line Encyclopedia of Integer Sequences."
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