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Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(F(k)*a(k)) < 1, F = Fibonacci.
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%I #15 Oct 20 2024 14:14:52

%S 2,3,4,9,44,1486,1357976,1855074754595,2975714380792664939835466,

%T 46528348836004781630107949818181021469921360198769

%N Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(F(k)*a(k)) < 1, F = Fibonacci.

%H Alois P. Heinz, <a href="/A377229/b377229.txt">Table of n, a(n) for n = 1..14</a>

%e s(0), s(1), ... = 0, 1/2, 5/6, 23/24, 215/216, 11879/11880, 17653679/17653680, ... .

%p F:= combinat[fibonacci]:

%p s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+1/(F(n)*a(n))) end:

%p a:= proc(n) option remember; 1+floor(1/((1-s(n-1))*F(n))) end:

%p seq(a(n), n=1..11);

%Y Cf. A000045, A079586, A374663, A376058, A377205, A377230.

%K nonn

%O 1,1

%A _Alois P. Heinz_, Oct 20 2024