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A374404 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1). For n > 2 if rad(i*j) is primorial number A002110(t), a(n) is least novel k such that rad(i*j*k) = A002110(t+1). Otherwise, if rad(i*j) is not primorial, a(n) is least novel k such that rad(i*j*k) = A002110(s), where prime(s) = A006530(i*j); rad = A007947.

%I A374404 #11 Jul 19 2024 19:12:11
%S A374404 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 A374404 140,13,66,105,26,44,210,39,55,42,52,110,63,65,88,84,40,77,36,45,49,
%U A374404 16,60,56,99,50,98,48,75,112,121,90,126,132,80,147,143,100,168
%N A374404 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1). For n > 2 if rad(i*j) is primorial number A002110(t), a(n) is least novel k such that rad(i*j*k) = A002110(t+1). Otherwise, if rad(i*j) is not primorial, a(n) is least novel k such that rad(i*j*k) = A002110(s), where prime(s) = A006530(i*j); rad = A007947.
%C A374404 Definition is subtly different from those of A362889 and A374379, which share the same initial terms of this sequence (divergence at a(53) = 98). If rad(i*j) is primorial = A002110(t), a(n) is least novel prime(t+1)-smooth number divisible by prime(t+1). And if rad(i*j) is not primorial, a(n) is the least novel prime(s)-smooth multiple of A002110(s)/rad(i*j). a(n) is prime iff rad(i*j) is a primorial number not seen earlier as kernel of the product of any prior pair of consecutive terms. It follows from the definition that for any consecutive three terms i,j,k, rad(i*j*k) is always a primorial number.
%C A374404 Conjectured to be a permutation of the positive integers A000027, with primes in order.
%H A374404 Michael De Vlieger, <a href="/A374404/b374404.txt">Table of n, a(n) for n = 1..10000</a>
%H A374404 Michael De Vlieger, <a href="/A374404/a374404_1.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple representing powerful numbers that are not prime powers.
%t A374404 nn = 120; c[_] := False; m[_] := 1;
%t A374404 Do[Set[{a[n], c[n], m[n]}, {n, True, 2}], {n, 3}]; i = a[2]; j = a[3];
%t A374404 f[x_] := f[x] = FactorInteger[x][[All, 1]];
%t A374404 q[x_] := Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ f[x] == {1}]];
%t A374404 Do[If[q[i*j],
%t A374404   s = NextPrime@Last@f[i*j]; k = 1;
%t A374404     While[Or[c[k*s], ! q[i*j*k*s]], k++]; k *= s,
%t A374404   t = Product[Prime[r], {r, PrimePi@ Last@ f[i*j]}];
%t A374404     s = t/Apply[Times, f[i*j]]; k = 1;
%t A374404     While[Or[c[k*s], Times @@ f[i*j*k*s] != t], k++]; k *= s];
%t A374404   Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, 4, nn}];
%t A374404 Array[a, nn] (* _Michael De Vlieger_, Jul 12 2024 *)
%Y A374404 Cf. A000027, A002110, A006530, A007947, A055932, A362889, A374379.
%K A374404 nonn
%O A374404 1,2
%A A374404 _David James Sycamore_ and _Michael De Vlieger_, Jul 07 2024