Sylvester's sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one.
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Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions with the same number of terms. Further, the sum of the first k terms of the infinite series of reciprocals provides the closest possible underestimate of 1 by any k-term Egyptian fraction.
- Task
- Write a routine (function, procedure, generator, whatever) to calculate Sylvester's sequence.
- Use that routine to show the values of the first 10 elements in the sequence.
- Show the sum of the reciprocals of the first 10 elements on the sequence;
- See also
Haskell
<lang haskell>sylvester :: [Integer] sylvester = map s [0..]
where s 0 = 2 s n = product (map s [0..n-1]) + 1
main :: IO () main = do
putStrLn "First 10 elements of Sylvester's sequence:" putStr $ unlines $ map show $ take 10 sylvester putStr "Sum of reciprocals: " putStrLn $ show $ sum $ map ((1/) . fromInteger) $ take 10 sylvester</lang>
- Output:
First 10 elements of Sylvester's sequence: 2 3 7 43 1807 3263443 10650056950807 113423713055421844361000443 12864938683278671740537145998360961546653259485195807 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 Sum of reciprocals: 0.9999999999999999
Raku
<lang perl6>my @S = {1 + [*] @S[^($++)]} … *;
put 'First 10 elements of Sylvester\'s sequence: ', @S[^10];
say "\nSum of the reciprocals of first 10 elements: ", sum @S[^10].map: { FatRat.new: 1, $_ };</lang>
- Output:
First 10 elements of Sylvester's sequence: 2 3 7 43 1807 3263443 10650056950807 113423713055421844361000443 12864938683278671740537145998360961546653259485195807 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 Sum of the reciprocals of first 10 elements: 0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999635