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A064442
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Decimal expansion of number with continued fraction expansion 2, 3, 5, 7, 11, 13, 17, 19, ... = 2.3130367364335829063839516 ...
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16
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2, 3, 1, 3, 0, 3, 6, 7, 3, 6, 4, 3, 3, 5, 8, 2, 9, 0, 6, 3, 8, 3, 9, 5, 1, 6, 0, 2, 6, 4, 1, 7, 8, 2, 4, 7, 6, 3, 9, 6, 6, 8, 9, 7, 7, 1, 8, 0, 3, 2, 5, 6, 3, 4, 0, 2, 1, 0, 1, 2, 4, 4, 4, 2, 1, 4, 4, 5, 6, 4, 7, 3, 1, 7, 7, 6, 2, 7, 2, 2, 4, 3, 6, 9, 5, 3, 2, 2, 0, 1, 7, 2, 3, 8, 3, 2, 8, 1, 7, 4, 5, 3, 0, 1, 5, 8, 2
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OFFSET
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1,1
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COMMENTS
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Continued fraction expansion of the prime numbers. - Harvey P. Dale, Sep 25 2012
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LINKS
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FORMULA
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Denoting the constant by c we have the related simple continued fraction expansions (prime(n) denotes the n-th prime number):
2*c = [4; 1, 1, 1, 2, 14, 5, 1, 1, 6, 34, 9, 1, 1, 11, 58, 15, 1, 1, 18, 82, 21, ..., 1, 1, (prime(3*n) - 1)/2, 2*prime(3*n+1), (prime(3*n+2) - 1)/2, ...];
(1/2)*c = [1; 6, 2, 1, 1, 3, 22, 6, 1, 1, 8, 38, 11, 1, 1, 14, 62, 18, 1, 1, 20, 86, 23, ..., 1, 1, (prime(3*n+1) - 1)/2, 2*prime(3*n+2), (prime(3*n+3) - 1)/2, ...];
(c + 1)/(c - 1) = [2; 1, 1, 10, 3, 1, 1, 5, 26, 8, 1, 1, 9, 46, 14, 1, 1, 15, 74, 20, ..., 1, 1, (prime(3*n+2) - 1)/2, 2*prime(3*n+3), (prime(3*n+4) - 1)/2, ...]. (End)
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EXAMPLE
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2.313036736433582906383951602641782476396689771803256340210124442144564731776...
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MATHEMATICA
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RealDigits[ N[ FromContinuedFraction[ Table[ Prime[n], {n, 1, 100} ]], 100]] [[1]]
RealDigits[FromContinuedFraction[Prime[Range[200]]], 10, 120][[1]] (* Harvey P. Dale, Sep 06 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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