Theodorus's Constant Digits

The digits of Theodorus's constant form the decimal expansion

(OEIS A002194). It was computed to decimal digits by E. Weisstein on Jul. 23, 2013.

The Earls sequence for gives the starting position of consecutive copies of the decimal representation of , counting the first digit after the decimal point as position 1. Its known first values for , 2, ... are 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874).

The known -constant primes occur at prefix lengths 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344), counting the integer-part 1 as the first digit.

The starting positions of the first occurrence of the decimal representation 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 , including the integer-part 1, until all -digit strings have occurred, including those with leading zeros, the reported last 1-, 2-, ...-digit strings are 4, 91, 184, 5566, 86134, 035343, ..., which end at digits 23, 378, 7862, 77437, 1237533, 16362668, ....

The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does, starting at positions 1104282392, 1879095207, 3037917993, ... (E. Weisstein, Jul. 23, 2013).

It is not known if is normal (Beyer et al. 1969, 1970ab), and the following table gives the reported counts of digits in the first fractional decimal digits, excluding the integer-part 1.

	OEIS	10	100
0		3	15	95	1035	10125	100234	1000172	9995281	99976638	1000006042
1		0	7	97	996	10019	99587	1001548	10001670	99988551	999978902
2		1	8	100	994	9829	99812	1000263	10001751	99991487	999982296
3		1	9	97	945	9898	99818	998943	10000247	100004464	999998469
4		0	7	84	971	10077	99897	998647	10001384	100023203	1000009144
5		2	13	93	1009	10037	100260	999993	9995879	99996674	999982506
6		0	10	103	1027	10052	100558	999976	9999931	100020148	1000025094
7		2	11	98	991	9921	99921	1000059	10002655	99987934	999997927
8		1	14	125	1002	9996	100055	1000650	10001042	100017107	1000013674
9		0	6	108	1030	10046	99858	999749	10000160	99993794	1000005946