코플랜드-에르되시 상수(Copeland-Erdős Constant)는 아서 허버트 코플랜드와 폴 에르되시(Paul Erdös)가 함께 작업한 상수이다.
소수를 이용하여 정의한 유사 정규수(normal number)이다. 코플랜드와 에르되시는 이 상수가 10 진법에 기초한 경우에서 정규수라는 것을 보여 주었다.[1][2][3]
![{\displaystyle C_{CE}=\sum _{n=1}^{\infty }{P_{n}}10^{\left(-\sum _{k=1}^{n}\lceil {log_{10}^{(1+P_{k})}}\rceil \right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aebf32f081e82589ad091c5b5b73d221d18b284a)
![{\displaystyle \;\;\;=\sum _{n=1}^{\infty }{P_{n}}10^{-\left(n+\sum _{k=1}^{n}\lfloor {log_{10}^{P_{k}}}\rfloor \right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/801cabf4f97b11832819cddb8c2f23c147b1488d)
![{\displaystyle \;\;\;=\sum _{n=1}^{\infty }{{P_{n}} \over {10^{\sum _{k=1}^{n}\lfloor log_{10}P_{k}\rfloor +n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dee3e4e27943cb27fc88c18fe3c44c1c0295ebc8)
소수와 관련하여 비교적인 측면에서 챔퍼나운 수(Champernowne constant)의 연분수에는 산발적인 매우 큰 주기(large term 또는 long term)가 포함되어 있기 때문에 연분수를 계산하기가 어려워지지만 코플랜드-에르되시 상수(Copeland-Erdős Constant)의 연분수는 잘 작동하면서 "롱텀(long term)현상"을 나타내지도 않는다.[4][5][6]
![{\displaystyle 2,3,5,7,1,1,1,3,1,7,1,9,2,3,2,9,3,1,3,7,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e606b688c9746ae0d7df0576c49d6c392067afdc)
![{\displaystyle 4,1,4,3,4,7,5,3,5,9,6,1,6,7,7,1,7,3,7,9,8,3,8,9,9,7,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f917b455728d86d84ff6878bbf54941f0575f7d)
![{\displaystyle 1,0,1,1,0,3,1,0,7,1,0,9,1,1,3,1,2,7,1,3,1,1,3,7,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0cf380c8e92ac1376a8746491c6eac45be7823)
![{\displaystyle 1,3,9,1,4,9,1,5,1,1,5,7,1,6,3,1,6,7,1,7,3,1,7,9,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5eaac769420b0cfe5f7c4f9dcef8fa433dc9be)
(A033308OEIS)[8]
![{\displaystyle 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a58446ecdcb3d97df757c00336d79448b90ffe0)
![{\displaystyle ,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d98ce75c86bc7d64e8176b56ecdbd8fa5d0f28e9)
![{\displaystyle ,151,157,163,167,173,179,181,191,193,197,199,211,223,227,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a00f6058e5a7555da36f83a621513e626356d13)
![{\displaystyle 229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d64a1f7b0395b8c73e7a46e54f545be15d00ca8)
![{\displaystyle 313,317,331,337,347,349,353,359,367,373,379,383,389,397,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb06c4e4eb33f467a22411bd381f7bccb3b448d)
![{\displaystyle 401,409,419,421,431,433,439,443,449,457,461,463,467,479,....}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99ea38024b43029bcbf2f91e031abdaac874ea35)
- Sloane, N. J. A. Sequences A019518, A030168, A033308, A033309, A033310, and A224890 in "The On-Line Encyclopedia of Integer Sequences."
- ↑ Copeland, A. H. and Erdős, P. "Note on Normal Numbers." Bull. Amer. Math. Soc. 52, 857-860, 1946.
- ↑ (OEIS)http://oeis.org/A033308
- ↑ (OEIS)http://oeis.org/A068670 (Daniel Forgues, Apr 02 2014)
- ↑ Allouche, J.-P. and Shallit, J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, p. 334, 2003.
- ↑ Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.
- ↑ Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." J. London Math. Soc. 8, 1933.
- ↑ (A033308 as a constant ,usually base 10)0.23571113171923293137414347535961677173798389971011031071091131271311371391491511571631671731791811911 (https://oeis.org/A033308/constant)
- ↑ (OEIS)http://oeis.org/A033308