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Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 10/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.
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%I #4 May 01 2013 12:24:43

%S 1,9,819,7519239,695384944860879,6470289227069622272847335347359,

%T 605164280025029017271801950447677089988237937249820002811725119

%N Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 10/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

%C Numerators of the sequence s(n) of the sum resp. product of fractions f(n) is A165428(n+2), hence sum(A165428(i+1)/A225162(i),i=1..n) = product(A165428(i+1)/A225162(i),i=1..n) = A165428(n+2)/a(n) = A220812(n-1)/a(n).

%F a(n) = 10^(2^(n-1))*b(n) where b(n)=b(n-1)-b(n-1)^2 with b(1)=1/10.

%e f(n) = 10, 10/9, 100/91, 10000/9181, ...

%e 10 + 10/9 = 10 * 10/9 = 100/9; 10 + 10/9 + 100/91 = 10 * 10/9 * 100/91 = 10000/819; ...

%e s(n) = 1/b(n) = 10, 100/9, 10000/819, ...

%p b:=proc(n) option remember; b(n-1)-b(n-1)^2; end:

%p b(1):=1/10;

%p a:=n->10^(2^(n-1))*b(n);

%p seq(a(i),i=1..7);

%Y Cf. A076628, A165428, A220812, A225162.

%K nonn

%O 1,2

%A _Martin Renner_, Apr 30 2013