Soldner's Constant Digits

Ramanujan calculated  (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is

(OEIS A070769; Derbyshire 2004, p. 114). The first  decimal digits were computed by E. Weisstein on Oct. 7, 2013.

-constant primes occur for 4, 144, 227, 444, 19474, ... (OEIS A122422) decimal digits.

The Earls sequence (starting position of  copies of the digit ) for  is given for , 2, ... by 3, 42, 178, 10013, 31567, 600035, 1253449, ... (OEIS A229071).

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of  (not including the initial 0 to the left of the decimal point) are 17, 1, 8, 5, 2, 3, 6, 34, 11, ... (OEIS A229201).

Scanning the decimal expansion of  until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 465, 102, 5858, 48441, ... (OEIS A000000), which end at digits 34, 512, 7454, 92508, 1414058, ... (OEIS A000000).

The digit sequences 0123456789 and 9876543210 do not occur in the first  digits (E. Weisstein, Oct. 7, 2013).

It is not known if  is normal, 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	0	9	116	1053	10098	100104	999785
1	A000000	1	7	100	970	9893	100238	1000370
2	A000000	1	9	106	979	10113	100057	999594
3	A000000	2	13	109	1012	10120	99999	1001006
4	A000000	2	10	96	1019	10118	99822	999546
5	A000000	1	15	103	994	9912	99918	1001007
6	A000000	1	8	89	1036	10060	99971	999430
7	A000000	0	6	97	988	10029	100141	997185
8	A000000	1	15	101	988	9838	100089	1001593
9	A000000	1	8	83	961	9819	99661	1000484

REFERENCES:

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 123-124, 1994.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 23 and 45, 1999.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 39, 1983.

Sloane, N. J. A. Sequences A070769, A122422, A229071, A229201 in "The On-Line Encyclopedia of Integer Sequences."