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%I
%S 1,8,1784,2322,2781,6133,6619,12814,15199,54262,70863,72751,208731,
%T 231730,273554,279748,422298,1821146,2439961,2655408,2748048,3022960,
%U 3174338,4582596,5001307,6350113,9137740,11650986
%N a(1)=1, a(n) is the smallest integer > a(n-1) such that the largest element in the simple continued fraction for S(n)=1/a(1)+1/a(2)+...+1/a(n) equals n^3.
%e The continued fraction for S(5)=1+1/8+1/1784+1/2322+1/2781 is {1, 7, 1, 10, 1, 2, 3, 1, 125, 1, 1, 6} where the largest element is 125=5^3 and 2781 is the smallest integer > 2322 with this property.
%t a[1] = 1; a[n_] := a[n] = Block[{s = Sum[1/a[i], {i, 1, n - 1}], k = a[n - 1] + 1}, While[ Max[ ContinuedFraction[s + 1/k]] != n^3, k++ ]; k]; Do[ Print[ a[n]], {n, 1, 28}]
%o (PARI) s=1; t=1; for(n=2,17,s=s+1/t; while(abs(n^3-vecmax(contfrac(s+1/t)))>0,t++); print1(t,","))
%K easy,nonn
%O 1,2
%A _Benoit Cloitre_, May 19 2002
%E Edited and extended by _Robert G. Wilson v_, Jun 10 2002
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