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علماء الرياضيات | Constant Digit Scanning |
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 05:51 مساءً
date: 26-1-2020 |
Author : المرجع الالكتروني للمعلوماتيه 
Book or Source : المرجع الالكتروني للمعلوماتيه 
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Date: 11-3-2020
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Scan the decimal expansion of a constant (including any digits to the left of the decimal point) until all 
-digit strings have been seen (including 0-padded strings). The following table then gives the number of digits that must be scanned to encounter all 
, 2, ...-digit strings (where "number of digits" means the ending-not starting-digit of an 
-digit string) together with the last 
-digit string encountered.
| constant | OEIS | sequence |
| Apéry's constant | A036906 | 23, 457, 7839, 83054, 1256587, 13881136, 166670757, ... |
| A036902 | 7, 89, 211, 2861, 43983, 29270, 8261623, ... | |
| Catalan's constant | A000000 | 32, 716, 7700, 86482, 1143572, ... |
| A000000 | 8, 45, 529, 2679, 24200, ... | |
| Champernowne constant | A072290 | 11, 192, 2893, 38894, 488895, 5888896, 68888897, 788888898, 8888888899, ... |
| Copeland-Erdős constant | A000000 | 48, 934, 24437, 366399, 4910479, 49672582, ... |
| A000000 | 0, 84, 504, 8580, 07010, 088880, ... | |
| e | A036904 | 21, 372, 8092, 102128, 1061613, 12108841, 198150341, 1929504534, ... |
| A036900 | 6, 12, 548, 1769, 92994, 513311, 1934715, 56891305, ... | |
| Euler-Mascheroni constant | A000000 | 16, 658, 6600, 91101, 1384372, ... |
| A000000 | 8, 18, 346, 2778, 84514, ... | |
| Glaisher-Kinkelin constant | A000000 | 22, 495, 7233, ... |
| A000000 | 5, 98, 478, ... | |
| golden ratio | A000000 | 23, 770, 5819, 93910, 1154766, 13192647, ... |
| A000000 | 5, 55, 515, 0092, 67799, 290503, ... | |
| Golomb-Dickman constant | A000000 | 28, 587, 6322, ... |
| A000000 | 1, 33, 821, ... | |
| Khinchin's constant | A000000 | 23, 499, 8254, ... |
| A000000 | 7, 43, 782, ... | |
| natural logarithm of 2 | A036905 | 22, 444, 7655, 98370, 1107795, 12983306, ... |
| A036901 | 2, 98, 604, 1155, 46847, 175403, ... | |
| natural logarithm of 10 | A229124 | 22, 701, 7486, 88092, 1189434, 13426407, ... |
| A229126 | 7, 38, 351, 8493, 33058, 362945, ... | |
| pi | A080597 | 33, 607, 8556, 99850, 1369565, 14118313, 166100507, 1816743913, 22445207407, 241641121049, 2512258603208, ... |
| A032510 | 0, 68, 483, 6716, 33394, 569540, 1075656, 36432643, 172484538, 5918289042, 56377726040, ... | |
| Pythagoras's constant | A000000 | 19, 420, 8326, 94388, 1256460, 13043524, ... |
| A000000 | 8, 81, 748, 8505, 30103, 489568, ... | |
| Soldner's constant | A000000 | 34, 512, 7454, 92508, ... |
| A000000 | 7, 46, 102, 5858, ... | |
| Theodorus's constant | A000000 | 23, 378, 7862, 77437, 1237533, 16362668, ... |
| A000000 | 4, 91, 184, 5566, 86134, 35343, ... |
The starting positions of the first occurrence of 
, 1, 2, ... in the decimal expansion of a number of constants are summarized in the table below, where any initial 0 to the left of the decimal point is ignored non any nonzero initial digits are counted as the "first" digit.
| constant | OEIS | first occurrence of 0, 1, 2, ... |
| Apéry's constant | A229187 | 3, 1, 2, 10, 16, 6, 7, 23, 18, 8, ... |
| Catalan's constant | A100079 | 16, 2, 13, 24, 9, 3, 5, 11, 32, 1, ... |
| Champernowne constant | A229186 | 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... |
| Copeland-Erdős constant | A229190 | 48, 5, 1, 2, 21, 3, 31, 4, 41, 12, ... |
| e | A088576 | 14, 3, 1, 18, 11, 12, 21, 2, 4, 13, ... |
| Euler-Mascheroni constant | A229192 | 11, 5, 4, 14, 9, 1, 7, 2, 16, 10, 36, ... |
| Glaisher-Kinkelin constant | A229193 | 12, 1, 2, 18, 5, 22, 14, 7, 3, 10, 11, ... |
| Golomb-Dickman constant | A229195 | 15, 28, 2, 4, 3, 10, 1, 17, 8, 6, 28, ... |
| golden ratio | A088577 | 5, 1, 20, 6, 12, 23, 2, 11, 4, 8, 232, ... |
| Khinchin's constant | A229196 | 8, 10, 1, 14, 5, 4, 2, 23, 3, 22, 10, ... |
| natural logarithm of 2 | A100077 | 9, 4, 22, 3, 5, 10, 1, 6, 8, 2, 108, ... |
| natural logarithm of 10 | A229197 | 3, 21, 1, 2, 13, 5, 17, 22, 6, 9, 41, ... |
| pi | A032445 | 33, 2, 7, 1, 3, 5, 8, 14, 12, 6, 50, ... |
| Pythagoras's constant | A229199 | 14, 1, 5, 7, 2, 8, 9, 12, 19, 15, 77, ... |
| Soldner's constant | A229201 | 17, 1, 8, 5, 2, 3, 6, 34, 11, 7, 16, ... |
| Theodorus's constant | A229200 | 5, 1, 4, 3, 23, 6, 12, 2, 8, 18, 48, ... |
REFERENCES:
Sloane, N. J. A. Sequences A032445, A032510, A036900, A036901, A036902, A036904, A036905, A036906, A072290, A080597, A088576, A088577, A100077, A100079, A229124, A229126, A229186, A229187, A229190, A229192, A229193, A229195, A229197, A229199, A229200, and A229201 in "The On-Line Encyclopedia of Integer Sequences."
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