Natural Logarithm of 10 Digits

The numerical value of  is given by

(OEIS A002392). It was computed to  decimal digits by S. Kondo on May 20, 2011 (Yee).

The Earls sequence (starting position of  copies of the digit ) for  is given for , 2, ... by 20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082, ... (OEIS A228243).

-constant primes occur at 1, 2, 40, 242, 842, 1541, 75067, ... decimal digits (OEIS A228240).

The starting positions of the first occurrence of , 1, ... in the decimal expansion of  (including the initial 2 and counting it as the first digit) are 3, 21, 1, 2, 13, 5, 17, 22, ... (OEIS A229197).

Scanning the decimal expansion of  until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 22, 701, 7486, 88092, 1189434, 13426407, ... (OEIS A229124), which end at digits 7, 38, 351, 8493, 33058, 362945, ... (OEIS A229126).

The digit strings 0123456789 first occurs starting at position 3349545080, but 9876543210 does not occur in the first  decimal digits of  (E. Weisstein, Aug. 27, 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	2	14	103	964	9979	99887	1000356	9996731	99987842	1000052045
1	A000000	0	5	82	1039	10088	100429	998340	9999245	100008060	1000031412
2	A000000	2	11	106	1008	9943	99927	1000879	10005133	100013939	1000008327
3	A000000	1	8	81	1045	10108	100166	999329	10001639	100010112	999996738
4	A000000	0	9	106	973	10006	100210	998402	9999948	99988853	999996641
5	A000000	2	9	103	967	9961	99375	1001381	10000670	100011829	999956649
6	A000000	0	9	107	967	9828	99769	999489	10001182	99972292	999972639
7	A000000	0	11	110	1007	10061	99782	1000327	9999623	99989099	999983529
8	A000000	1	10	106	1000	9987	100633	1001812	9998524	100004534	999974468
9	A000000	2	14	96	1030	10039	99822	999685	9997305	100013440	1000027552

REFERENCES:

Sloane, N. J. A. Sequences A002392/M0394, A228240,A228243, and A229197 in "The On-Line Encyclopedia of Integer Sequences."

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