cp's OEIS Frontend

A354755 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.

%I A354755 #18 Jun 16 2022 07:20:28
%S A354755 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 A354755 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 A354755 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 A354755 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 A354755 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 A354755 Michael De Vlieger, <a href="/A354755/b354755.txt">Table of n, a(n) for n = 1..10000</a>
%H A354755 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 A354755 Scott R. Shannon, <a href="/A354755/a354755.png">Image of the first 500000 terms</a>. The green line is y = n.
%e A354755 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 A354755 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 A354755 (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 A354755 Cf. A064413, A354727, A354687, A354753, A353989, A354087, A352763
%K A354755 nonn,look
%O A354755 1,2
%A A354755 _Scott R. Shannon_, Jun 06 2022