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Earls Sequence

المؤلف:  Pickover, C. A.

المصدر:  The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press

الجزء والصفحة:  ...

27-1-2020

1273

Earls Sequence

 

The Earls sequence gives the starting position in the decimal digits of  (or in general, any constant), not counting digits to the left of the decimal point, at which a string of  copies of the number  first occurs. The following table gives generalized Earls sequences for various constants, including .

constant	OEIS	sequence
Apéry's constant	A229074	10, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ...
Catalan's constant	A224819	2, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ...
Champernowne constant	A224896	1, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ...
Copeland-Erdős constant	A224897	5, 113, 1181, 21670, 263423, 7815547, 35619942, 402720247, 450680638
e	A224828	2, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011
Euler-Mascheroni constant	A224826	5, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186
Glaisher-Kinkelin constant	A225763	7, 14, 2264, 1179, 411556, ...
golden ratio	A224844	2, 62, 158, 1216, 72618, 2905357, 7446157, 41398949, 1574998166
Golomb-Dickman constant	A225242	28, 256, 1967, 387, ...
Khinchin's constant	A224836	9, 42, 1799, 494, 5760, ...
natural logarithm of 2	A228242	4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693, 2539803904
natural logarithm of 10	A228243	20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082
pi	A061073	1, 135, 1698, 54525, 24466, 252499, 3346228, 46663520, 564665206
Pythagoras's constant	A224871	2, 114, 1481, 3308, 72459, 226697, 969836, 119555442, 2971094743
Soldner's constant	A229071	3, 42, 178, 10013, 31567, ...
Theodorus's constant	A224874	27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423

Note that the 20-digit sequence  does not occur in the first  digits of , nor in fact do any of the -digit sequences  for any two-digit number  with  and  (E. W. Weisstein, Jul. 20, 2013).

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REFERENCES:

Anderson, D. "The Pi-Search Page." http://www.angio.net/pi/piquery.

Pickover, C. A. The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, p. 338, 2002.

Sloane, N. J. A. Sequences A061073, A224819, A224826, A224828, A224836, A224844, A224871, A224874, A224896, A224897, A225242, A225763, A228242, A228243, A229074, and A229071 in "The On-Line Encyclopedia of Integer Sequences."