Pythagoras's Constant Digits

Pythagoras's constant  has decimal expansion

(OEIS A000129), It was computed to  decimal digits by A. J. Yee on Feb. 9, 2012.

The Earls sequence (starting position of  copies of the digit ) for  is given for , 2, ... by 2, 114, 1481, 3308, 72459, 226697, 969836, 119555442, 2971094743, ... (OEIS A224871).

-constant primes occur at 55, 97, 225, 11260, 11540, ... (OEIS A115377) decimal digits.

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of  (including the initial 1 and counting it as the first digit) are 14, 1, 5, 7, 2, 8, 9, 12, 19, ... (OEIS A229199).

Scanning the decimal expansion of  until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 81, 748, 8505, 30103, 489568, ... (OEIS A000000), which end at digits 19, 420, 8326, 94388, 1256460, 13043524, ... (OEIS A000000).

The digit sequence 9876543210 does not occur in the first  digits of , but 0123456789 does, starting at positions 864106288, 6458611884, 7311432557, ... (OEIS A000000) (E. Weisstein, Jul. 22, 2013).

It is not known if  is normal (Beyer et al. 1969, 1970ab), 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	10	108	952	9959	99814	999897	10002237	100010228	999996989
1	A000000	2	7	98	1005	10106	98924	1000114	10000179	99998381	1000042849
2	A000000	2	8	109	1004	9876	100436	1000208	9998091	99995645	999987069
3	A000000	2	11	82	980	10058	100191	999674	10004178	99995415	999984900
4	A000000	2	9	100	1016	10100	100024	1000126	10000054	100012725	1000008724
5	A000000	1	7	104	1001	10002	100155	999358	9998344	100002636	999970045
6	A000000	1	10	90	1032	9939	99886	1001246	10001665	100012683	1000007824
7	A000000	0	18	104	964	10008	100008	999359	9998646	99980315	999986743
8	A000000	0	12	113	1027	10007	100441	999452	9996550	99995120	1000025363
9	A000000	0	8	92	1019	9945	100121	1000566	10000056	99996852	999989494

REFERENCES:

Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent." Math. Comput. 23, 679, 1969.

Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Statistical Study of Digits of Some Square Roots of Integers in Various Bases." Math. Comput. 24, 455-473, 1970a.

Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases." Math. Comput. 24, 745-747, 1970b.

Sloane, N. J. A. Sequences A000129/M1314, A115377, A224871, and A229199 in "The On-Line Encyclopedia of Integer Sequences."

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