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a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) * a(n-2) + |a(n-1) - a(n-2)|.
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%I #24 May 02 2022 12:40:29

%S 1,2,3,7,5,37,14,541,8101,223,23,73,13,1009,11,12097,46,22,4,6,8,10,

%T 12,16,18,15,9,21,24,26,20,28,30,32,34,25,859,35,17,613,69,42841,39,

%U 1713601,19,92,27,2549,38,43,33,1429,115,44,42,36,40,48,50,52,54,45,51,57,60,49,65,55,63

%N a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) * a(n-2) + |a(n-1) - a(n-2)|.

%C The sequence produces numerous groupings of primes. For example a(3) to a(16) contains thirteen primes in fourteen terms, a(80) to a(102) contains fourteen primes in twenty-three terms. The sequences is conjectured to be a permutation of the positive integers.

%H Michael De Vlieger, <a href="/A353075/b353075.txt">Table of n, a(n) for n = 1..5000</a>

%H Michael De Vlieger, <a href="/A353075/a353075.png">Log-log scatterplot of a(n)</a>, n = 1..5000, showing primes in red.

%e a(5) = 5 as a(4)*a(3)+|a(4)-a(3)| = 7*3+|7-3| = 25, and 5 is the smallest unused number that shares a factor with 25.

%t nn = 69; c[_] = 0; a[1] = c[1] = 1; a[2] = c[2] = 2; u = 3; Do[m = #1 #2 + Abs[#2 - #1] & @@ {a[i - 2], a[i - 1]}; If[PrimeQ[m], k = 1; While[c[k m] != 0, k++]; k = k m, k = u; While[Nand[c[k] == 0, ! CoprimeQ[m, k]], k++]]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, May 02 2022 *)

%Y Cf. A353006, A353082, A352867, A352774, A352768.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Apr 22 2022