OFFSET
1,2
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.
LINKS
Scott R. Shannon, Table of n, a(n) for n = 1..10000.
EXAMPLE
a(6) = 12 as |12 - 3| = 9, and 9 is a divisor of 12*3 = 36. No smaller unused number has this property.
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Sep 29 2023
STATUS
approved