Catalan's Constant Digits

Based on methods developer in collaboration with M. Leclert, Catalan (1865) computed the constant

(OEIS A006752) now known as Catalans' constant to 9 decimals. In 1867, M. Bresse subsequently computed  to 24 decimal places using a technique from Kummer. Glaisher evaluated  to 20 (Glaisher 1877) and subsequently 32 decimal digits (Glaisher 1913). Catalan's constant was computed to  decimal digits by A. Roberts on Dec. 13, 2010 (Yee).

The Earls sequence (starting position of  copies of the digit ) for Catalan's constant is given for , 2, ... by 2, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ... (OEIS A224819).

-constant primes occur for 52, 276, 25477, ... (OEIS A118328) digits.

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	A224615	0	6	98	976	9828	99620	999784	9998686	99996067
1	A224616	2	18	94	1039	9832	99697	1000293	10003813	100006305
2	A224696	0	10	93	980	10078	100168	1001789	10005122	100000806
3	A224706	0	7	104	1014	9859	99580	999672	9995676	100001483
4	A224717	1	11	107	961	10051	100074	1000165	9995377	100001871
5	A224774	3	10	89	1003	10062	100053	999965	9999309	100000777
6	A224775	1	12	78	985	9986	100201	998712	10000674	99998816
7	A224816	0	11	124	1032	10028	100083	1000510	10003863	100000576
8	A224817	0	3	102	1058	10192	100352	999298	9997437	100000863
9	A224818	3	12	111	952	10084	100172	999812	10000043	99992436

The digits 0123456789 do not occur in the first  decimal digits of , but 9876543210 does (once), starting at position 2748123761 (E. Weisstein, Aug. 7, 2013).

REFERENCES:

Glaisher, J. W. L. "On a Numerical Continued Product." Messenger Math. 6, 71-76, 1877.

Glaisher, J. W. L. "Numerical Values of the Series  for , 4, 6." Messenger Math. 42, 35-58, 1913.

Sloane, N. J. A. Sequences A118328 and A224819 in "The On-Line Encyclopedia of Integer Sequences."

Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/.