A373944 Lexicographically earliest sequence of distinct positive integers such that for A(k) <= n < A(k+1); rad(Product_{i = 1..n} a(i)) = A002110(k), where A = A002110, rad = A007947, k >= 0, n >= 1.

1, 2, 4, 8, 16, 3, 6, 9, 12, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 5, 10, 15, 20, 25, 30, 40, 45, 50, 60, 75, 80, 90, 100, 120, 125, 135, 150, 160, 180, 200, 225, 240, 250, 270, 300, 320, 324, 360, 375, 384, 400

Offset: 1

Author

David James Sycamore, Jun 23 2024

Keywords

• nonn
• easy

Comments

Sequence is computed piecewise in blocks of A002110(k+1) - A002110(k) terms, for indices n in the range A002110(k) <= n < A002110(k+1), k = 0,1,2,... in which all terms are the ordered earliest prime(k)-smooth numbers not already recorded in earlier blocks. Since a(0) = 1, and for all k >= 1, all prime(k)-smooth numbers eventually appear in the sequence, this is a permutation of the positive integers, A000027.

From Michael De Vlieger, Jun 25 2024: (Start)

Let P(i) = A002110(i) be the product of i smallest primes.

Let rad = A007947 and let gpf = A006530.

Let S(i) = {k : rad(k) | P(i)}, the prime(i)-smooth numbers.

The notation S(i,j) denotes the j-th smallest term in i, i.e., the j-th term when S(i) is sorted.

This sequence can be seen as a table with row r = 0 {1}, r = 1 {2, 4, 8, 16}, etc.

Then row r contains k in S(r, 1..P(r+1)-1) such that terms k <= S(r-1, P(r)-1) such that gpf(k) < prime(r) are removed.

As a consequence, the sorted union of rows 0..r reconstructs S(r, 1..P(r+1)-1).

For example, A003586(1..29) is given by the sorted union of rows r = 0..2 of the sequence.

The sorted union of rows r = 0..3 gives A051037(1..209), etc.

For r > 1, P(r) is the P(r-1)-th term in row r. (End)

Examples

			k = 0 --> A(0) <= n < A(1) --> 1 <= n < 2 --> n = 1 --> a(1) = 1 since rad(1) = 1 = A(0).
k = 1 --> A(1) <= n < A(2) --> 2 <= n < 6 --> n = 2,3,4,5 --> a(2,3,4,5) = 2,4,8,16 (the first 4 terms of A000079, excluding 1).
k = 2 --> 6 <= n < 30 --> n = 6,7,8,9,...,29 --> a(6,7,8,9...,29) = 3,6,9,12,...,288 (the first 24 terms of A003586 excluding all above).
k = 3 --> 30 <= n < 210 --> n = 30,31,32,...,209 --> a(30,31,32,...,209) = 5,10,15,...,19200 (the first 180 terms of A051037 excluding all above).
Sequence can be presented as an irregular table T(n,k), in which the n-th row commences A008578(n); n >= 1, and T(n,k) is the k-th prime(n)-smooth number which has not appeared earlier.
Table starts:
  1;
  2,4,8,16;
  3,6,9,12,18,24,27,32,...,288;
  5,10,15,20,25,30,40,45,50,60,...,19200;
  7,14,21,28,...,13829760;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n) n = 1..510509.

Crossrefs

Cf. A000040, A002110, A002473, A003586, A007947, A051037, A051038, A080197, A080681, A080682, A080683.

Programs

Mathematica

(* First, load function f from A162306 *)
P = m = 1; Flatten@ Join[{{1}}, Reap[Do[P *= Prime[i]; (Sow@ Select[#, Nand[# <= m, FactorInteger[#][[-1, 1]] < Prime[i]] &]; m = #[[-1]]) &@ f[P, P^4][[;; P*Prime[i + 1] - 1]], {i, 3}] ][[-1, 1]]] (* Michael De Vlieger, Jun 24 2024 *)

Formula

a(A002110(n)) = A000040(n), n >= 1.