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%I #12 Jun 17 2022 03:29:39
%S 1,2,2,4,4,6,3,9,6,8,8,10,10,12,9,15,10,16,12,15,15,18,14,20,15,21,18,
%T 20,20,22,22,24,21,27,24,26,26,28,21,35,20,38,19,57,3,63,7,56,6,58,10,
%U 55,20,52,22,55,25,60,9,69,12,70,14,72,15,75,18,70,21,75,24,68,26,72,28,74,30,65
%N a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and both the sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=2..n-1, and the product a(n) * a(n-1) is distinct from all previous products, a(i) * a(i-1), i=2..n-1
%C This sequence uses a combination of the term selection rules of A354755 and A354753. The first forty-five terms are the same as A354755 beyond which they differ; see the examples below. In the first 500000 terms only six terms are prime, 2,3,7,19, with 2 and 3 occurring twice, the last being a(47) = 7. It is unknown if more appear. The only fixed points are 1,2,4,6, and it is likely no more exist.
%H Scott R. Shannon, <a href="/A354858/a354858.png">Image of the first 500000 terms</a>. The green line is y = n.
%e a(7) = 3 as a(6) = 6, and 3 is the smallest number that shares a factor with 6 and whose sum and product with the previous term, 6 + 3 = 9 and 6 * 3 = 18, have not previously appeared. Note 2 shares a factor with 6 but 6 + 2 = 8, and a sum of 8 has already occurred with a(4) + a(5) = 4 + 4 = 8, so 2 cannot be chosen.
%e a(46) = 63 as a(45) = 3, and 63 is the smallest number that shares a factor with 3 and whose sum and product with the previous term, 3 + 63 = 66 and 3 * 63 = 189, have not previously appeared. Note 60 shares a factor with 3 but the product 3 * 60 = 180 has already occurred with a(19) * a(20) = 12 * 15 = 180, so 60 cannot be chosen. This is the first term to differ from A354755.
%o (PARI) lista(nn) = my(va = vector(nn), vp = vector(nn-2), vs = vector(nn-2)); va[1] = 1; va[2] = 2; for (n=3, nn, my(k=2); while ((gcd(k, va[n-1]) == 1) || #select(x->(x==k*va[n-1]), vp) || #select(x->(x==k+va[n-1]), vs), k++); va[n] = k; vp[n-2] = k*va[n-1]; vs[n-2] = k+va[n-1];); va; \\ _Michel Marcus_, Jun 17 2022
%Y Cf. A354755, A354753, A088177, A064413, A354803.
%K nonn,look
%O 1,2
%A _Scott R. Shannon_, Jun 09 2022