1,2

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.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

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.

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.

(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

Cf. A354755, A354753, A088177, A064413, A354803.

Sequence in context: A354753 A377225 A354755 * A128923 A153835 A338098

Adjacent sequences: A354855 A354856 A354857 * A354859 A354860 A354861

nonn,look

Scott R. Shannon, Jun 09 2022

approved