A376244 - OEIS

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 #6 Sep 16 2024 12:36:20

%S 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 22,37

%N 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 Is this sequence infinite?

%H Rémy Sigrist, <a href="/A376244/a376244.gp.txt">PARI program</a>

%e The initial terms are:

%e n a(n) b(n) S(n)

%e - ---- ------ ---------------------------

%e 1 3 1 1/3

%e 2 4 1 7/12

%e 3 5 1 47/60

%e 4 4 2 109/120

%e 5 7 2 823/840

%e 6 3 17 4757/4760

%e 7 9 177 7582661/7582680

%e 8 1 399089 3026164178509/3026164178520

%o (PARI) \\ See Links section.

%Y Cf. A374663, A376062, A376184, A376245 (corresponding b's), A376246-A376247 (numerator and denominator of corresponding S(n)).

%K nonn,more

%O 1,1

%A _Rémy Sigrist_ and _N. J. A. Sloane_, Sep 16 2024