Theodorus's Constant Digits

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."