cp's OEIS Frontend

A354753 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 product a(n) * a(n-1) is distinct from all previous products a(i) * a(i-1), i=2..n-1.

%I A354753 #28 Oct 25 2022 11:38:24
%S A354753 1,2,2,4,4,6,2,10,4,8,6,3,3,9,6,6,10,5,5,15,3,21,6,12,8,8,10,10,12,9,
%T A354753 9,15,6,14,2,22,4,14,7,7,21,9,18,8,14,10,15,12,14,14,16,8,20,10,22,6,
%U A354753 26,2,34,4,26,8,22,11,11,33,3,39,6,32,8,30,9,24,12,21,14,20,15,15,21,18,18
%N A354753 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 product a(n) * a(n-1) is distinct from all previous products a(i) * a(i-1), i=2..n-1.
%C A354753 This sequence uses a similar rule to A088177 but here all neighboring terms also share a factor. In the first 500000 terms the fixed points are 1,2,4,6, it is likely no more exist, while the smallest number not to have appeared is 1153. The sequence is conjectured to be a permutation of the positive integers.
%C A354753 See A354754 for the products of all pairs of terms.
%H A354753 Michael De Vlieger, <a href="/A354753/b354753.txt">Table of n, a(n) for n = 1..10000</a>
%H A354753 Michael De Vlieger, <a href="/A354753/a354753_1.png">Annotated log-log scatterplot of a(n)</a> n = 1..2^14, showing records in red, a(n) = 2 in blue, fixed points highlighted in gold.
%H A354753 Scott R. Shannon, <a href="/A354753/a354753.png">Image of the first 50000 terms</a>.
%e A354753 a(7) = 2 as a(6) = 6 and 2 is the smallest positive number that shares a factor with 6 and whose product with 6, 2 * 6 = 12, has not previously appeared.
%t A354753 nn = 120; c[_] = 0; a[1] = c[1] = 1; a[2] = a[2] = j = 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 A354753 (PARI) lista(nn) = my(va = vector(nn), vp = 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), k++); va[n] = k; vp[n-2] = k*va[n-1];); va; \\ _Michel Marcus_, Jun 14 2022
%Y A354753 Cf. A354754, A354803, A354804, A354749, A354759, A088177, A064413.
%K A354753 nonn,look
%O A354753 1,2
%A A354753 _Scott R. Shannon_, Jun 06 2022