Khinchin's Constant Digits

The numerical value of Khinchin's constant  is given by

(OEIS A002210). However, the numerical value of  is notoriously difficult to calculate to high precision. Bailey et al. (1997) computed  to 7350 digits, and the current record is  digits, computed by Xavier Gourdon in 1997 with a computation requiring 22 hours and 23 minutes (Plouffe).

The Earls sequence (starting position of  copies of the digit ) for Khinchin's constant is given for , 2, ... by 9, 42, 1799, 494, 5760, ... (OEIS A224836), with the  term being larger than .

-constant primes occur at 1, 407, 878, 4443, 4981, 6551, 13386, 28433, ... decimal digits (OEIS A118327).

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of  (including the initial 2 and counting it as the first digit) are 8, 10, 1, 14, 5, 4, 2, 23, 3, 22, ... (OEIS A229196).

Scanning the decimal expansion of  until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 43, 782, ... (OEIS A000000), which end at digits 23, 499, 8254, ... (OEIS A000000).

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	3	10	101	1030	9991
1	A000000	1	12	95	1004	10070
2	A000000	1	9	98	967	9890
3	A000000	0	10	105	1039	9840
4	A000000	1	11	96	977	9943
5	A000000	2	15	92	1045	10116
6	A000000	1	6	107	999	10106
7	A000000	0	7	103	953	10020
8	A000000	1	9	95	970	9942
9	A000000	0	11	108	1016	10082

REFERENCES:

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khintchine Constant." Math. Comput. 66, 417-431, 1997.

Plouffe, S. "Table of Current Records for the Computation of Constants." http://pi.lacim.uqam.ca/eng/records_en.html.

Plouffe, S. "New Record on the Computation of the Digits of the Khintchine Constant." http://pi.lacim.uqam.ca/piDATA/khintchine.txt.

Sloane, N. J. A. Sequences A002211/M1564 and A118327 in "The On-Line Encyclopedia of Integer Sequences."a