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%I #18 Jun 16 2022 07:20:28
%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 20,20,22,22,24,21,27,24,26,26,28,21,35,20,38,19,57,3,60,2,62,4,64,6,
%U 63,9,66,8,70,7,77,11,88,2,78,3,84,2,80,5,60,32,62,31,93,3,99,6,92,8,96,10,85,25
%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 the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.
%C In the first 500000 terms the fixed points are 1,2,4,6,2388,2390,2392,2394; it is likely no more exist. In the same range many numbers do not appear, the lowest five being 59,67,73,89,97. It is possible these and many other numbers never appear although this is unknown.
%H Michael De Vlieger, <a href="/A354755/b354755.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A354755/a354755_1.png">Annotated log-log scatterplot of a(n)</a> n = 1..2^14, showing records in red and a(n) = 2 in blue, highlighting fixed points in gold.
%H Scott R. Shannon, <a href="/A354755/a354755.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 with the previous term, 6 + 3 = 9, has not 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.
%t nn = 120; c[_] = 0; a[1] = c[1] = 1; a[2] = j = 2; c[3] = 2; Do[k = 2; While[Nand[c[j + k] == 0, ! CoprimeQ[j, k]], k++]; Set[{a[n], c[j + k]}, {k, n}]; j = k, {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, Jun 15 2022 *)
%o (PARI) lista(nn) = my(va = vector(nn), 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]), vs), k++); va[n] = k; vs[n-2] = k+va[n-1];); va; \\ _Michel Marcus_, Jun 15 2022
%Y Cf. A064413, A354727, A354687, A354753, A353989, A354087, A352763
%K nonn,look
%O 1,2
%A _Scott R. Shannon_, Jun 06 2022
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