OFFSET
1,2
COMMENTS
Conjecture: limsup_{n->infinity} a(n) = infinity. More precisely we claim that log(a(n))/log(n) is bounded and does not converge to zero (see related link). Does a(n) = 2 infinitely many times or does it exist M >= 2 such that a(n) <= M infinitely many times?
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
1/a(1)+1/a(2)+1/a(3)+1/a(4)=1+1/2+1/3+1/11 = 127/66 and the continued fraction expansion is [1, 1, 12, 5] containing 4 elements. Next k=2 is the smallest integer >=1 such that the continued fraction expansion of 127/66+1/k which is [2, 2, 2, 1, 4] contains 5 elements, thus a(5) = 2.
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Module[{s = Total[1/Array[a, n - 1]], k = 2}, While[Length[ContinuedFraction[s + 1/k]] != n, k++]; k]; Array[a, 100] (* Amiram Eldar, Jun 05 2022 *)
PROG
(PARI) v=[1]; for(m=2, 100, k=1; while(abs(length(contfrac(1/k+sum(i=1, length(v), 1/v[i])))-m)>0, k++); v=concat(v, [k])); a(n)=v[n];
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jan 09 2013
STATUS
approved