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Search: id:A109366
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| A109366 |
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Sequence of denominators of the continued fraction derived likewise to the continued fraction found by Euler for e-1 but based on prime numbers only. |
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+0
2
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| 1, 2, 9, 55, 448, 5533, 77753, 1415862, 28378685, 685274581, 20695944714, 662817798145, 25290008485783
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OFFSET
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0,2
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COMMENT
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The value of the continued fraction (general formulation) up to n is: R(n) = A(n)/B(n) where: A(0) = 1 A(1) = a(1)*A(0) + b(1) A(n) = a(n)*A(n - 1) + b(n)*A(n-2) (n>=2) Euler form for (e-1)is ( e - 1) = 1+ (2 / (2 + 3 / (3 + 4/(4 + 5/ (5 + 6 /...), by using prime numbers only, we can define a constant ep so that (ep -1) = 1+ (2 / (2 + 3 / (3 + 5/(5 + 7/ (7 + 11 /...), ep = 2.71961651193526... > e. More precisely: ep/e = 1.00049100..
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FORMULA
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B(0) = 1, B(1) = a(1) *B(0), B(n) = a(n)* B(n - 1) + b(n)*B(n-2) (n>=2) where a(1) = P(1), a(2) = P(2) ..., a(n) = P(n) where P(n) the n-th prime number ( P(1) =2 ) and b(1) = P(1), b(2) = P(2) ..., b(n) = P(n)
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EXAMPLE
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n = 2: B(2)=9 because: B(0) = 1, B(1) = P(1)*B(0) = 1*2 = 2, B(2) = p(2)*B(1)+p(2)*B(0)=3*2 + 3*1=9
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CROSSREFS
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Cf. A109365.
Sequence in context: A036074 A009363 A069564 this_sequence A091108 A081004 A138740
Adjacent sequences: A109363 A109364 A109365 this_sequence A109367 A109368 A109369
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KEYWORD
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frac,nonn,uned
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AUTHOR
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Giorgio Balzarotti (greenblue(AT)tiscali.it), Aug 23 2005
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