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A381112 a(1) = 1, let q = greatest prime in S(n) = {p; p = A053669(a(i)); 1 <= i <= n-1}. Then for n > 1, a(n) is the smallest number not yet in the sequence such that: (i) q|a(n), and (ii) p a prime and p^k|a(n) implies p in S(n) and k <= cardinality of p in S(n).

%I A381112 #20 Mar 07 2025 09:21:57
%S A381112 1,2,3,6,5,10,15,20,30,7,14,21,28,35,42,56,63,70,84,105,112,126,140,
%T A381112 168,175,189,210,11,22,33,44,55,66,77,88,99,110,132,154,165,176,198,
%U A381112 220,231,264,275,297,308,330,352,385,396,440,462,495,528,539,550,594
%N A381112 a(1) = 1, let q = greatest prime in S(n) = {p; p = A053669(a(i)); 1 <= i <= n-1}. Then for n > 1, a(n) is the smallest number not yet in the sequence such that: (i) q|a(n), and (ii) p a prime and p^k|a(n) implies p in S(n) and k <= cardinality of p in S(n).
%C A381112 Sequence contains all primes (in order), but no perfect prime powers. All primorial numbers are terms but no multiples m*A002110(k) are terms, where m = 2,3,4,...,prime(k+1)-1. Each prime p results in a trajectory (see graph) of p-smooth numbers corresponding to rows of a table; see Example.
%C A381112 Let P(i) = A002110(i). For n > 178 and a(n) = P(i)/prime(i-1), a(n+1) = prime(i-1)^2 * prime(i). See tables in Example. - _Michael De Vlieger_, Mar 06 2025
%H A381112 Michael De Vlieger, <a href="/A381112/b381112.txt">Table of n, a(n) for n = 1..6382</a>
%H A381112 Michael De Vlieger, <a href="/A381112/a381112.png">Log log scatterplot of a(n)</a>, n = 1..6382.
%H A381112 Michael De Vlieger, <a href="/A381112/a381112_1.png">Plot prime(i)^m | a(n) at (x,y) = (n,i)</a>, n = 1..4096, 24X vertical exaggeration, with a color function representing m = 1 in black, m = 2 in red, ..., largest m in the dataset in magenta. The colored bar below the image is red if a(n) is prime, gold if a(n) is a proper prime power, green for squarefree composite a(n), and blue or purple if a(n) is neither squarefree nor prime power, where purple additionally represents powerful a(n) that are not prime powers.
%H A381112 Michael De Vlieger, <a href="/A381112/a381112.txt">Notes on the mechanics of this sequence</a>, regarding the prime decomposition of a(n) and the cardinalities of primes p in the sequence S(1..n), where S(n) = A053669(a(n)).
%e A381112 The smallest nondivisor prime of a(1) = 1 is 2, so S(2) = {2} and so a(2) = 2. Likewise the smallest nondivisor prime of a(2) = 2 is 3, so S(3) = {2,3} and a(3) = 3. At this point we have S(4) = {2,3,2} and the next term must have greatest prime factor = q = 3 and other prime factors must be in S(4). Therefore a(4) = 2*3 = 6.
%e A381112 Sequence expressed in tabular form, rows (after first) starting with a prime and ending with a primorial:
%e A381112   1;
%e A381112   2;
%e A381112   3,6;
%e A381112   5,10,15,20,30;
%e A381112   7,14,21,28,35,42,56,63,70,84,105,112,126,140,168,175,189,210;
%e A381112   11,22,33,44,55,66,...
%e A381112 Row (n+1) commencing p = prime(n) contains all p-smooth numbers m; p <=m <= A002110(n) except for any perfect powers p^k, (k>1) of p which lie in that range.
%t A381112 (* First, load the program regs from the "fast Mathematica programs" in the links at A369609, then: *)
%t A381112 nn = 8; p = q = 1; c[_] := False; m[_] := 0;
%t A381112 f[x_] := f[x] = Block[{qx = 2}, While[Divisible[x, qx], qx = NextPrime[qx]]; qx];
%t A381112 {1}~Join~Reap[Do[
%t A381112   Map[If[! c[#],
%t A381112     If[PrimeQ[#],
%t A381112       Sow[#]; c[#] = True; m[f[#]]++,
%t A381112         If[And[AllTrue[FactorInteger[#], #2 <= m[#1] & @@ # &],  Divisible[#, p]],
%t A381112           Sow[#]; c[#] = True; m[f[#]]++] ] ] &, regs[q, q]];
%t A381112   q *= Prime[i]; p = Prime[i]; u = 1, {i, nn}] ][[-1, 1]] (* _Michael De Vlieger_, Feb 16 2025 *)
%Y A381112 Cf. A000040, A002110, A006530, A053669.
%K A381112 nonn
%O A381112 1,2
%A A381112 _David James Sycamore_, Feb 14 2025