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A368133 a(1,2,3) = 1,2,3; let j = a(n-1), M(n) = Product_{i = 1..n-2} { p a distinct prime: p | a(i), gcd(p, j) = 1 }. For n > 3, a(n) is the least novel multiple of M(n) if M(n) > 1; otherwise a(n) is the least novel multiple of A053669(j), the smallest prime which does not divide j.

%I A368133 #27 Jun 21 2025 07:53:50
%S A368133 1,2,3,4,6,5,12,10,9,20,15,8,30,7,60,14,45,28,75,42,25,84,35,18,70,21,
%T A368133 40,63,50,105,16,210,11,420,22,315,44,525,66,140,33,280,99,350,132,
%U A368133 175,198,245,264,385,24,770,27,1540,36,1155,26,2310,13,4620,39,3080
%N A368133 a(1,2,3) = 1,2,3; let j = a(n-1), M(n) = Product_{i = 1..n-2} { p a distinct prime: p | a(i), gcd(p, j) = 1 }. For n > 3, a(n) is the least novel multiple of M(n) if M(n) > 1; otherwise a(n) is the least novel multiple of A053669(j), the smallest prime which does not divide j.
%C A368133 M(n) is a squarefree number whose prime factors are the distinct primes which divide a(m), m <= n-2, but do not divide j. M(n) > 1 implies there exists at least one term prior to j having a prime divisor which does not divide j, and M(n) is the product of all such primes. If, for any term a(m), m <= n-2, every prime factor of a(m) also divides j, then M(n) = 1, the empty product.
%C A368133 Primorial a(n-1) implies prime a(n); see Formula.
%C A368133 Conjectured to be a permutation of the positive integers.
%C A368133 Compare with A368108 which has a slightly different definition but works in a similar way.
%C A368133 From _Michael De Vlieger_, Jan 05 2024: (Start)
%C A368133 This sequence is the same as A362855 for 91306 terms.
%C A368133 A362855(91306) = a(91306) = A002110(17),
%C A368133 A362855(91307) = 53 = prime(16), a(91307) = 61 = prime(18),
%C A368133 A362855(91308) = A002110(17)/prime(16), a(91308) = 2*A002110(17).
%C A368133 Thereafter the sequences diverge. It seems unlikely that the 2 sequences will become coincident again as n increases beyond 91308. (End)
%H A368133 Michael De Vlieger, <a href="/A368133/a368133.png">Log log scatterplot of a(n) and b(n)</a>, n = 1..2^20, where a(n) is shown in red, b(n) = A362855(n) is shown in blue. Black indicates where a(n) = b(n).
%F A368133 When for some m, a(m) = A002110(n), a primorial number, a(m+1) = prime(n+1), a(m+2) = 2*A002110(n), and a(m+3) = 2*prime(n+1); see Example.
%F A368133 a(n) = A362855(n), for 1 <= n <= 91306 (see link and Example).
%e A368133 a(1, 2, 3) = 1, 2, 3. M(4) = 2 because 2 | a(2) but does not divide a(3); 2 is the only a(m), m < 3, with this property, so a(4) = 4, the least novel multiple of 2.
%e A368133 Now we have a(1,2,3,4) = 1,2,3,4. M(5) = 3 because 3 | a(3) but does not divide a(4); 3 is the only a(m), m < 4, with this property, so a(5) = 2*3 = 6, the least novel multiple of 3.
%e A368133 We now have a(1..5) = 1, 2, 3, 4, 6. M(6) = 1, the empty product, because there is no prime which divides some a(m), m < 5, which does not also divide a(n-1) = 6. This situation invokes the second condition of the definition, so a(6) = 1*5, the least novel multiple of A053669(6) = 5, the smallest prime which does not divide 6. Consequently a(7) = 2*6 = 12 because no prime dividing a(1..5) also divides 5.
%e A368133 The same situation arises again at a(13) = 30 = 2*3*5; every prime divisor of a(m), m < 13, is 2, 3, or 5, which again invokes the second condition, M(14) = 1, the empty product, so a(14) = 1*7, since A053669(30) = 7. Consequently a(15) = 2*7 = 14.
%e A368133 a(91307) = 61 (whereas A362855(91307) = 53; point of divergence from A362855).
%t A368133 nn = 10^5; c[_] := False; m[_] := 1;
%t A368133 Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 3]; j = 3;
%t A368133 s = {2}; r = Max[s]; c[3] = False;
%t A368133 q[x_] := Block[{qq = 2}, While[Divisible[x, qq], qq = NextPrime[qq]]; qq];
%t A368133 Do[(If[# == 1,
%t A368133         Set[k, NextPrime[r]],
%t A368133         While[Or[c[#], # == j] &[# m[#]], m[#]++];
%t A368133           Set[k, # m[#]]] &[Times @@ Complement[s, #]];
%t A368133           s = Union[s, #];
%t A368133      If[Last[#] > r, r = Last[#]]) &@ FactorInteger[j][[All, 1]];
%t A368133   Set[{a[n], c[j], j}, {k, True, k}], {n, 4, nn}];
%t A368133 Array[a, nn] (* _Michael De Vlieger_, Jan 05 2024 *)
%Y A368133 Cf. A002110, A053669, A351495, A362855, A368108.
%K A368133 nonn
%O A368133 1,2
%A A368133 _David James Sycamore_, Dec 13 2023
%E A368133 More terms from _Michael De Vlieger_, Jan 05 2024