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.

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, 84, 35, 40, 32, 34, 17, 306, 102, 51, 42, 33, 22, 11, 132, 44, 46, 23, 552, 138, 69, 60, 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

Offset: 1

Author

Scott R. Shannon, Sep 29 2023

Keywords

• nonn

Comments

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).

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.

Examples

			a(6) = 12 as |12 - 3| = 9, and 9 is a divisor of 12*3 = 36. No smaller unused number has this property.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000.

Crossrefs

Cf. A027750, A363576, A359799.