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A376244 Lexicographically earliest sequence of positive integers a(1), a(2), ... with the property that the lexicographically earliest sequence of positive integers b(1), b(2), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1 / (a(k)*b(k)) < 1, also implies that S(n) is never of the form (e_n - 1) / e_n for some integer e_n.

%I A376244 #6 Sep 16 2024 12:36:20
%S A376244 3,4,5,4,7,3,9,1,11,4,13,7,9,19,10,2,23,25,29,27,53,1,17,7,2,2,15,67,
%T A376244 22,37
%N A376244 Lexicographically earliest sequence of positive integers a(1), a(2), ... with the property that the lexicographically earliest sequence of positive integers b(1), b(2), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1 / (a(k)*b(k)) < 1, also implies that S(n) is never of the form (e_n - 1) / e_n for some integer e_n.
%C A376244 Is this sequence infinite?
%H A376244 Rémy Sigrist, <a href="/A376244/a376244.gp.txt">PARI program</a>
%e A376244 The initial terms are:
%e A376244   n  a(n)  b(n)    S(n)
%e A376244   -  ----  ------  ---------------------------
%e A376244   1     3       1  1/3
%e A376244   2     4       1  7/12
%e A376244   3     5       1  47/60
%e A376244   4     4       2  109/120
%e A376244   5     7       2  823/840
%e A376244   6     3      17  4757/4760
%e A376244   7     9     177  7582661/7582680
%e A376244   8     1  399089  3026164178509/3026164178520
%o A376244 (PARI) \\ See Links section.
%Y A376244 Cf. A374663, A376062, A376184, A376245 (corresponding b's), A376246-A376247 (numerator and denominator of corresponding S(n)).
%K A376244 nonn,more
%O A376244 1,1
%A A376244 _Rémy Sigrist_ and _N. J. A. Sloane_, Sep 16 2024