Notes on A113552. Michael De Vlieger, St. Louis, Missouri 201805181800. "Beginning with 1, least divisor of the previous term not included earlier, otherwise the least multiple of the previous term having at least one prime divisor coprime to it and not included earlier." In the second condition, we are looking for a number that does not divide some positive integer power e of n (i.e., "non-regular" to n). A regular number m with respect to n is one such that m | n^e with e >= 0. The divisor is a special case of n-regular number. In the table below, we note a cycle that subtends for 24 <= n <= 2^14. Let e = floor(n/12). We write k if the condition is false, or the parity of d if d does not occur in a. We can express a(n) as the product of the smallest four primes as shown below. n (mod 12) k or d 2 3 5 7 ------------------------------------------- 0 ODD 3^(e-1) 5 7 1 6 2 3^e 5 7 2 ODD 3^e 3 2 2 3^e 4 5 2 3^e 5 5 ODD 3^e 5 6 4 2^2 3^e 5 7 EVEN 2^2 3^e 8 7 2^2 3^e 7 9 ODD 3^e 7 10 2 2 3^e 7 11 10 2^2 3^e 5 7 Let's look at the first terms of a(n): The table below shows a(n) for 1 <= n <= 36. We write multiple k if the condition is false, or "0" if d does not occur in a. We can express a(n) as the product of the smallest four primes as shown below, noting multiplicities of the prime divisors of a(n). n k or d 2.3.5.7 a(n) ----------------------------- 1 0 0 1 2 2 1 2 3 3 1.1 6 4 0 0.1 3 5 4 2.1 12 6 0 2 4 7 5 2.0.1 20 8 0 0.0.1 5 9 2 1.0.1 10 10 3 1.1.1 30 11 0 0.1.1 15 12 4 2.1.1 60 13 7 2.1.1.1 420 14 0 0.0.0.1 7 15 2 1.0.0.1 14 16 3 1.1.0.1 42 17 0 0.1.0.1 21 18 4 2.1.0.1 84 19 0 2.0.0.1 28 20 5 2.0.1.1 140 21 0 0.0.1.1 35 22 2 1.0.1.1 70 23 3 1.1.1.1 210 24 0 0.1.1.1 105 25 6 1.2.1.1 630 26 0 0.2 9 27 2 1.2 18 28 5 1.2.1 90 29 0 0.2.1 45 30 4 2.2.1 180 31 0 2.2 36 32 7 2.2.0.1 252 33 0 0.2.0.1 63 34 2 1.2.0.1 126 35 10 2.2.1.1 1260 36 0 0.2.1.1 315 ... The pattern thus continues as described above. Conjectures: 1. All terms are divisible only by some combination of the smallest 4 primes; A113553 is finite at four terms. 2. For n > 24 such that n (mod 12) = 2, a(n) = 3^((n - 2)/12). (eof)