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A366111 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared such that |a(n) - a(n-1)| is a divisor of a(n)*a(n-1), and where |a(n) - a(n-1)| > 1.

%I A366111 #13 Oct 04 2023 06:49:20
%S A366111 1,2,4,6,3,12,8,10,5,30,15,18,9,36,20,16,14,7,56,24,21,28,26,13,182,
%T A366111 84,35,40,32,34,17,306,102,51,42,33,22,11,132,44,46,23,552,138,69,60,
%U A366111 45,48,39,52,50,25,150,75,66,54,27,108,72,63,70,65,78,74,37,1406,684,171,90,80,55,110,85
%N A366111 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared such that |a(n) - a(n-1)| is a divisor of a(n)*a(n-1), and where |a(n) - a(n-1)| > 1.
%C A366111 Many of the terms lie just above the line a(n) = n, although this is not true of the prime-valued terms. Any prime factor of the difference |a(n) - a(n-1)| must be a factor of both a(n) and a(n-1), therefore if a term p is prime then the other term is a multiple of that prime, a*p. By the definition of the sequence a(n)*a(n-1) = a*p^2 must be a multiple of a*p - p = p*(a-1). This can only be true if a = 2 or a = p+1, thus the difference between the terms must be p or p^2. As a prime p cannot appear as a term if it has not previously appeared as a factor of a term, if a term is prime then the previous term must be 2*p and the following term must be p+p^2. Thus primed-valued terms force the following term to be O(p^2).
%C A366111 In the first 10000 terms the fixed points are 16, 21, 48, 98, 105, 322, 3088, 7659, although more likely exist. The sequence is conjectured to be a permutation of the positive integers.
%H A366111 Scott R. Shannon, <a href="/A366111/b366111.txt">Table of n, a(n) for n = 1..10000</a>.
%e A366111 a(6) = 12 as |12 - 3| = 9, and 9 is a divisor of 12*3 = 36. No smaller unused number has this property.
%Y A366111 Cf. A027750, A363576, A359799.
%K A366111 nonn
%O A366111 1,2
%A A366111 _Scott R. Shannon_, Sep 29 2023