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Lexicographically earliest sequence of distinct positive numbers such that a(1) = 1, a(2) = 2, and for n > 2, a(n) divides Sum_{i = n-k..n-1} a(i) with k > 0 as small as possible.
4

%I #20 Aug 23 2024 15:04:48

%S 1,2,3,5,4,9,13,11,6,17,23,8,31,39,7,46,53,33,43,19,62,27,89,29,59,22,

%T 81,103,92,15,107,61,12,73,85,79,41,10,51,34,95,129,14,143,157,20,177,

%U 197,187,16,203,219,211,86,99,37,68,21,18,24,42,66,36,102

%N Lexicographically earliest sequence of distinct positive numbers such that a(1) = 1, a(2) = 2, and for n > 2, a(n) divides Sum_{i = n-k..n-1} a(i) with k > 0 as small as possible.

%C When computing a(n) for n > 2, there may be candidates for different values of k; we choose the candidate that minimizes k.

%C This sequence is an infinite variant of A085947; a(n) = A085947(n) for n = 1..39.

%H Rémy Sigrist, <a href="/A328444/b328444.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A328444/a328444.gp.txt">PARI program for A328444</a>

%F a(n) <= Sum_{k = 1..n-1} a(k) for any n > 2.

%e For n = 3:

%e - the divisors of a(2) = 2 are all already in the sequence,

%e - 3 is the least divisor of a(1) + a(2) = 1 + 2 = 3 not yet in the sequence,

%e - so a(3) = 3.

%e For n = 4:

%e - the divisors of a(3) = 3 are all already in the sequence,

%e - 5 is the least divisor of a(2) + a(3) = 2 + 3 = 5 not yet in the sequence,

%e - so a(3) = 5.

%e For n = 5:

%e - the divisors of a(4) = 5 are all already in the sequence,

%e - 4 is the least divisor of a(3) + a(4) = 3 + 5 = 8 not yet in the sequence,

%e - so a(5) = 4.

%o (PARI) \\ See Links section.

%Y See A328443 for a similar sequence.

%Y Cf. A085947.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Oct 15 2019