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).

1, 2, 3, 6, 5, 10, 15, 20, 30, 7, 14, 21, 28, 35, 42, 56, 63, 70, 84, 105, 112, 126, 140, 168, 175, 189, 210, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 132, 154, 165, 176, 198, 220, 231, 264, 275, 297, 308, 330, 352, 385, 396, 440, 462, 495, 528, 539, 550, 594

Offset: 1

Author

David James Sycamore, Feb 14 2025

Keywords

• nonn

Comments

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.

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

Examples

			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.
Sequence expressed in tabular form, rows (after first) starting with a prime and ending with a primorial:
  1;
  2;
  3,6;
  5,10,15,20,30;
  7,14,21,28,35,42,56,63,70,84,105,112,126,140,168,175,189,210;
  11,22,33,44,55,66,...
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..6382
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..6382.
• Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), 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.
• Michael De Vlieger, Notes on the mechanics of this sequence, 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)).

Crossrefs

Cf. A000040, A002110, A006530, A053669.

Programs

Mathematica

(* First, load the program regs from the "fast Mathematica programs" in the links at A369609, then: *)
nn = 8; p = q = 1; c[] := False; m[] := 0;
f[x_] := f[x] = Block[{qx = 2}, While[Divisible[x, qx], qx = NextPrime[qx]]; qx];
{1}~Join~Reap[Do[
  Map[If[! c[#],
    If[PrimeQ[#],
      Sow[#]; c[#] = True; m[f[#]]++,
        If[And[AllTrue[FactorInteger[#], #2 <= m[#1] & @@ # &],  Divisible[#, p]],
          Sow[#]; c[#] = True; m[f[#]]++] ] ] &, regs[q, q]];
  q *= Prime[i]; p = Prime[i]; u = 1, {i, nn}] ][[-1, 1]] (* Michael De Vlieger, Feb 16 2025 *)