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Largest Fibonacci number in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.
4

%I #23 Mar 31 2015 01:07:21

%S 0,1,3,5,13,13,13,89,89,89,1597,1597,1597,1597,1597,1597,17711,17711,

%T 17711,28657,28657,28657,28657,1346269,1346269,1346269,1346269,

%U 24157817,24157817,24157817,24157817,24157817,24157817,39088169,39088169,39088169,39088169

%N Largest Fibonacci number in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

%C The prime Fibonacci numbers in the sequence are 3, 5, 13, 89, 1597, 28657, ...

%C For information about how often the numerator of these sums is a Fibonacci number, see A256220 and A256221.

%H Hiroaki Yamanouchi, <a href="/A256222/b256222.txt">Table of n, a(n) for n = 0..50</a>

%e a(3) = 5 because we obtain the 5 subsets {1}, {1/2}, {1/3}, {1,1/2} and {1/2, 1/3} having 5 sums with Fibonacci numerators: 1, 1, 1, 1+1/2 = 3/2 and 1/2+1/3 = 5/6 => the greatest Fibonacci number is 5.

%t <<"DiscreteMath`Combinatorica`"; maxN=24; For[t={}; mx=0; i=0; n=0, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[IntegerQ[Sqrt[5*k^2+4]]||IntegerQ[Sqrt[5*k^2-4]], If[k>mx, t=s]; mx=Max[mx, k]]]; Print[mx]]

%Y Cf. A000045, A005478, A075226, A256220, A256221, A256223.

%K nonn

%O 0,3

%A _Michel Lagneau_, Mar 19 2015

%E Corrected and extended by _Alois P. Heinz_, Mar 25 2015

%E a(30)-a(36) from _Hiroaki Yamanouchi_, Mar 30 2015