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A364872
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Lexicographically earliest continued fraction which is its own unit fraction series.
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2
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OFFSET
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0,1
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COMMENTS
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Earliest infinite sequence {a0,a1,a2,a3,...} such that: a0+1/(a1+1/(a2+1/(a3+...))) = 1/a0 + 1/a1 + 1/a2 + 1/a3 + ...
There are infinitely many real numbers whose continued fraction is also their unit fraction series - they are dense on the interval (2,oo).
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LINKS
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EXAMPLE
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The partial continued fraction must always be strictly larger than the partial unit fractions:
[1] cannot be since 1 = 1.
[2] can be since 2 > 1/2.
[2,1] can be since 2+1/1 > 1/2+1/1.
[2,1,1] cannot be since 2+1/(1+1/1) = 1/2+1/1+1/1.
[2,1,2] can be since 2+1/(1+1/2) > 1/2+1/1+1/2.
...
sum(1/a[n]) = 2.71053359137351078733864566... (A364873).
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PROG
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(PARI)
cf(a) = my(m=contfracpnqn(a)); m[1, 1]/m[2, 1];
uf(a) = sum(i=1, #a, 1/a[i]);
A364872(N) = {a=[2]; for(i=2, N, a=concat(a, if(cf(a)==uf(a), a[i-1], ceil(1/(cf(a)-uf(a))))); while(cf(a)<=uf(a), a[i]++)); a};
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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STATUS
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approved
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