%I #12 Nov 14 2022 00:34:53

%S 1,2,3,6,4,5,10,8,7,14,12,9,11,33,15,20,16,13,26,18,21,28,22,44,17,34,

%T 24,27,19,57,30,25,23,115,35,42,32,29,58,36,39,52,38,76,31,62,40,45,

%U 48,46,69,51,68,50,55,66,54,37,74,56,49,41,287,63,60,64,43

%N Let G(n) = gcd(a(n-2),a(n-1)), a(1)=1, a(2)=2, a(3)=3. Thereafter if G(n) = 1, a(n) is the least novel m sharing a divisor with both a(n-2) and a(n-1). If G(n) > 1 and every prime divisor of a(n-1) also divides a(n-2), a(n) is the least m prime to both a(n-1) and a(n-2). Otherwise a(n) is the least novel multiple of any prime divisor of a(n-1) which does not divide a(n-2).

%C Conjectured to be a permutation of the positive integers with the primes in natural order, and primes are the slowest numbers to appear (as in A352187).

%H Michael De Vlieger, <a href="/A358026/b358026.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A358026/a358026.png">Log-log scatterplot of a(n)</a>, n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and other prime powers in gold.

%e a(4) = 6, the least novel number sharing a factor with both 2 and 3.

%e a(5) = 4, the least novel multiple of 2, which divides a(4) but does not divide a(3).

%e Since every prime dividing a(5)=4 also divides a(4)=6, a(6)=5 the least novel term prime to 3 and 6.

%t nn = 67; c[_] = False; q[_] = 1; u = 4; Do[(Set[{a[n], c[n]}, {n, True}]; q[n]++), {n, u - 1}]; Do[m = FactorInteger[a[n - 1]][[All, 1]]; f = Select[m, CoprimeQ[#, a[n - 2]] &]; Which[Length[f] == PrimeNu[a[n - 1]], Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, Flatten@ Outer[Times, m, FactorInteger[a[n - 2]][[All, 1]] ] ], Last], Length[f] == 0, k = u; While[Nand[! c[k], CoprimeQ[a[n - 2], k], CoprimeQ[a[n - 1], k]], k++]; If[k == u, While[c[u], u++]], True, Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, f], Last] ]; Set[{a[n], c[k]}, {k, True}], {n, 4, nn}]; Array[a, nn] (* _Michael De Vlieger_, Oct 25 2022 *)

%Y Cf. A064413, A336957, A352187, A357963.

%K nonn

%O 1,2

%A _David James Sycamore_, Oct 25 2022

%E More terms from _Michael De Vlieger_, Oct 25 2022