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A362889 a(n) = n for n <= 2. Let i = a(n-2), j = a(n-1), q = gpd(i*j) = prime(s), and k = product of all distinct primes < q which do not divide i*j. For n > 2 a(n) is the least novel multiple of either k (if k is not the empty product), or of prime(s+1) if it is.

%I A362889 #12 Jun 23 2023 07:51:11
%S A362889 1,2,3,5,4,6,10,7,9,20,14,12,15,21,8,25,18,28,30,11,35,24,22,70,27,33,
%T A362889 140,13,66,105,26,44,210,39,55,42,52,110,63,65,88,84,40,77,36,45,49,
%U A362889 16,60,56,99,50,91,132,75,98,121,90,112,143,120
%N A362889 a(n) = n for n <= 2. Let i = a(n-2), j = a(n-1), q = gpd(i*j) = prime(s), and k = product of all distinct primes < q which do not divide i*j. For n > 2 a(n) is the least novel multiple of either k (if k is not the empty product), or of prime(s+1) if it is.
%C A362889 In other words, if rad(i*j) is a primorial number A002110(s) then there are no primes < q which do not divide i*j (k = empty product), and a(n) is the least novel multiple of the smallest prime > q. Otherwise there are distinct primes < q which do not divide i*j, and a(n) is the least novel multiple of their product.
%C A362889 a(n) is prime iff the radical of the product of the two previous terms is a primorial number which has not occurred previously; see Example.
%C A362889 The sequence grows rapidly due to occurrences of i*j which differ greatly from primorial numbers.
%C A362889 Conjectured to be a permutation of the positive integers (but primes are not in order, e.g. 41 precedes 37).
%H A362889 Michael De Vlieger, <a href="/A362889/b362889.txt">Table of n, a(n) for n = 1..10000</a>
%H A362889 Michael De Vlieger, <a href="/A362889/a362889.png">Log log scatterplot of a(n)</a>, n = 1..256, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
%H A362889 Michael De Vlieger, <a href="/A362889/a362889_1.png">Log log scatterplot of log_10 a(n)</a>, n = 1..2^16.
%H A362889 Michael De Vlieger, <a href="/A362889/a362889_2.png">Plot prime(i) | a(n) at (x,y) = (n,i)</a> for n = 1..2^11, with a color function conveying multiplicity, where black = 1, red = 2, orange = 3, etc. The strip of color on the bottom of the image shows red if a(n) is prime, gold if composite prime power, green if squarefree composite, and blue if neither squarefree nor prime powers.
%H A362889 Michael De Vlieger, <a href="/A362889/a362889.txt">Notes regarding this sequence</a>
%e A362889 a(1,2) = 1, 2;  i*j = 2 = A002110(1), so a(3) = prime(2) = 3.
%e A362889 a(2,3) = 2,3; i*j = 6 = A002110(2) so a(4) = prime(3) = 5.
%e A362889 a(3,4) = 3,5; and 2 is the only prime < 3 which does not divide 15, so a(5) = 4, the least novel multiple of 2.
%e A362889 a(4,5) = 5,4; and 3 is the only prime < 5 which does not divide rad(20) = 10, so a(6) = 6, the least novel multiple of 3.
%e A362889 a(17,18) = 18,28; and 5 is the only prime not dividing rad(504) = 42, thus a(19) = 30, the least novel multiple of 5.
%e A362889 a(18,19) = 28,30; and rad(28*30) = 210 = A002110(4), so a(20) = prime(5) = 11.
%e A362889 a(52,53) = 50,91; and rad(i*j) = 2*5*7*13 = 910 and 3,11 are the only primes  < 13 which do not divide 910 so a(54) is 132, the least novel multiple of 3*11 = 33.
%t A362889 mm = 3; nn = 2^15;
%t A362889   c[_] := False; m[_] := 1;
%t A362889   Array[Set[{a[#], c[#]}, {#, True}] &, mm]; m[2]++;
%t A362889   i = a[mm - 1]; j = a[mm]; q = FactorInteger[i j][[-1, 1]];
%t A362889   Do[t = PrimePi /@ FactorInteger[i j][[All, 1]]; q = Last[t];
%t A362889     s = DeleteCases[Range[q], _?(MemberQ[t, #] &)];
%t A362889     If[Length[s] == 0,
%t A362889       p = Prime[q + 1]; While[c[p*m[p]], m[p]++],
%t A362889       p = Times @@ Map[Prime, s]; While[c[p*m[p]], m[p]++]];
%t A362889     k = p*m[p]; Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, mm + 1, nn}];
%t A362889 Array[a, nn] (* _Michael De Vlieger_, May 08 2023 *)
%Y A362889 Cf. A002110, A007947.
%K A362889 nonn
%O A362889 1,2
%A A362889 _David James Sycamore_, May 08 2023