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A374379 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1), then for n > 2 a(n) is the least novel k such that rad(k) !| rad(i*j) whereas rad(i*j*k) is a primorial number (A002110), where rad is A007947.

%I A374379 #8 Aug 06 2024 21:51:39
%S A374379 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 A374379 140,13,66,105,26,44,210,39,55,42,52,110,63,65,88,84,40,77,36,45,49,
%U A374379 16,60,56,99,50,91,132,75,98,121,90,112,143,120,119,286,135,126,154,80,48,147
%N A374379 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1), then for n > 2 a(n) is the least novel k such that rad(k) !| rad(i*j) whereas rad(i*j*k) is a primorial number (A002110), where rad is A007947.
%C A374379 For n > 2 a(n) is prime iff rad(i*j) is a primorial number, for then k is the smallest prime which does not divide rad(i*j).
%C A374379 Initially same terms as A362889, but divergence occurs at a(69) = 147.
%C A374379 Conjectured to be a permutation of the natural numbers with primes in order.
%H A374379 Michael De Vlieger, <a href="/A374379/b374379.txt">Table of n, a(n) for n = 1..2050</a>
%H A374379 Michael De Vlieger, <a href="/A374379/a374379.png">Log log scatterplot of log_10(a(n))</a>, n = 1..2048.
%e A374379 a(3) = 3 since rad(3) !| rad(1*2) whereas rad (1*2*3) = 6 = A002110(2).
%e A374379 a(5,6) = {4,6}—> a(7) = 10 since rad(10) !| rad(24), rad(4*6*10) = 30 = A002110(3) and there is no smaller novel term with this property.
%e A374379 From _Michael De Vlieger_, Jul 06 2024: (Start)
%e A374379 Table of a(3..18) showing prime decomposition of rad(i*j) and rad(i*j*k):
%e A374379    n  a(n)   rad(i*j)   rad(i*j*k)
%e A374379   -------------------------------
%e A374379    3    3    2          2 3
%e A374379    4    5    2 3        2 3 5
%e A374379    5    4    . 3 5      2 3 5
%e A374379    6    6    2 . 5      2 3 5
%e A374379    7   10    2 3        2 3 5
%e A374379    8    7    2 3 5      2 3 5 7
%e A374379    9    9    2 . 5 7    2 3 5 7
%e A374379   10   20    . 3 . 7    2 3 5 7
%e A374379   11   14    2 3 5      2 3 5 7
%e A374379   12   12    2 . 5 7    2 3 5 7
%e A374379   13   15    2 3 . 7    2 3 5 7
%e A374379   14   21    2 3 5      2 3 5 7
%e A374379   15    8    . 3 5 7    2 3 5 7
%e A374379   16   25    2 3 . 7    2 3 5 7
%e A374379   17   18    2 . 5      2 3 5
%e A374379   18   28    2 3 5      2 3 5 7   (End)
%t A374379 nn = 1200; c[_] := False; rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
%t A374379 Array[Set[{a[#], c[#]}, {#, True}] &, 2];
%t A374379 f[n_] := Or[IntegerQ@ Log2[n],
%t A374379   And[EvenQ[n], Union@ Differences@ Map[PrimePi, FactorInteger[n][[All, 1]]] == {1}]]; i = a[1]; j = a[2]; u = 3;
%t A374379 Monitor[Do[r = rad[i*j]; k = u;
%t A374379   While[Or[c[k], Divisible[#, rad[k]], ! f[#  k]] &[i*j], k++];
%t A374379   Set[{a[n], c[k], i, j}, {k, True, j, k}];
%t A374379   If[k == u, While[c[u], u++]], {n, 3, nn}], n];
%t A374379 Array[a, nn] (* _Michael De Vlieger_, Jul 06 2024 *)
%Y A374379 Cf. A002110, A007947, A055932, A362889.
%K A374379 nonn
%O A374379 1,2
%A A374379 _David James Sycamore_, Jul 06 2024