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a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-1) + a(n) are factors of a(n).
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%I #29 Jan 07 2023 22:25:56

%S 1,2,6,3,24,8,56,42,7,336,48,16,112,84,12,4,28,21,60,15,10,22,66,30,

%T 18,9,72,36,45,80,20,5,120,40,85,204,39,78,26,38,90,35,14,50,75,150,

%U 93,186,57,114,102,34,94,162,54,27,216,108,135,240,144,99,198,44,77,266,95,380,132,110,11

%N a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-1) + a(n) are factors of a(n).

%C The primes do not occur in their natural order, and for all terms studied if a(n) is a prime p, then a(n-1) = p(p-1) and a(n+1) = p(p^2-1). In the first 10000 terms the fixed points are 22, 165, 710, 1005, 9003, although it is likely more exist. The sequence is conjectured to be a permutation of the positive integers.

%H Scott R. Shannon, <a href="/A359256/b359256.txt">Table of n, a(n) for n = 1..10000</a>.

%e a(3) = 6 as a(2) + 6 = 2 + 6 = 8 which has 2 as its only distinct prime factor, and 2 is a factor of 6.

%e a(8) = 42 as a(7) + 42 = 56 + 42 = 96 which has 2 and 3 as distinct prime factors, and 2 and 3 are factors of 42.

%e a(10) = 336 as a(9) + 336 = 7 + 336 = 343 which has 7 as its only distinct prime factor, and 7 is a factor of 336. Note that 336 = 7(7^2-1).

%Y Cf. A359557, A027748, A359356, A064413, A352867.

%K nonn

%O 1,2

%A _Scott R. Shannon_ and _Eric Angelini_, Jan 05 2023