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%I #10 Oct 10 2019 11:34:42
%S 1,3,8,10,2,7,4,5,6,9,11,12,13,14,15,16,17,18,19,20,22,21,23,24,25,27,
%T 26,28,29,30,31,32,33,34,35,36,37,38,39,40,43,41,42,44,45,46,47,48,49,
%U 50,51,52,53,54,55,57,56,58,59,60,61,63,62,64,65,66,67,68,71,69,70,72
%N a(n) is the smallest positive integer which does not occur earlier in the sequence and is such that sum{k=1 to n} 1/a(k) has a numerator which is composite (or is 1).
%C A127519(n)/A127520(n) is sum{k=1 to n} 1/a(k). This sequence, A127518, seems to be a permutation of the positive integers.
%e 1/a(1) + 1/a(2) +1/a(3) = 1 +1/3 + 1/8 = 35/24. 35/24 + 1/2 = 47/24; but 47 is prime, so a(4) is not 2. Likewise, an m of 4,5,6,7 and 9 give a numerator for (35/24 +1/m) which is prime. But 35/24 + 1/10 = 187/120. And since 187 = 11*17 is composite, then a(4) = 10.
%t f[l_List] := Sum[1/l[[k]], {k, Length[l]}];g[l_List] := Block[{k = 1, n},While[n = Numerator[f[Append[l, k]]]; MemberQ[l, k] || PrimeQ[n], k++ ];Append[l, k]];Nest[g, {}, 75] (* _Ray Chandler_, Jan 22 2007 *)
%Y Cf. A127515, A127519, A127520.
%K nonn
%O 1,2
%A _Leroy Quet_, Jan 17 2007
%E Extended by _Ray Chandler_, Jan 22 2007
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