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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.

%I A373944 #23 Jun 27 2025 08:34:37
%S A373944 1,2,4,8,16,3,6,9,12,18,24,27,32,36,48,54,64,72,81,96,108,128,144,162,
%T A373944 192,216,243,256,288,5,10,15,20,25,30,40,45,50,60,75,80,90,100,120,
%U A373944 125,135,150,160,180,200,225,240,250,270,300,320,324,360,375,384,400
%N 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.
%C A373944 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.
%C A373944 From _Michael De Vlieger_, Jun 25 2024: (Start)
%C A373944 Let P(i) = A002110(i) be the product of i smallest primes.
%C A373944 Let rad = A007947 and let gpf = A006530.
%C A373944 Let S(i) = {k : rad(k) | P(i)}, the prime(i)-smooth numbers.
%C A373944 The notation S(i,j) denotes the j-th smallest term in i, i.e., the j-th term when S(i) is sorted.
%C A373944 This sequence can be seen as a table with row r = 0 {1}, r = 1 {2, 4, 8, 16}, etc.
%C A373944 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.
%C A373944 As a consequence, the sorted union of rows 0..r reconstructs S(r, 1..P(r+1)-1).
%C A373944 For example, A003586(1..29) is given by the sorted union of rows r = 0..2 of the sequence.
%C A373944 The sorted union of rows r = 0..3 gives A051037(1..209), etc.
%C A373944 For r > 1, P(r) is the P(r-1)-th term in row r. (End)
%H A373944 Michael De Vlieger, <a href="/A373944/b373944.txt">Table of n, a(n) for n = 1..10000</a>
%H A373944 Michael De Vlieger, <a href="/A373944/a373944.png">Log log scatterplot of a(n)</a> n = 1..510509.
%F A373944 a(A002110(n)) = A000040(n), n >= 1.
%e A373944 k = 0 --> A(0) <= n < A(1) --> 1 <= n < 2 --> n = 1 --> a(1) = 1 since rad(1) = 1 = A(0).
%e A373944 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).
%e A373944 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).
%e A373944 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).
%e A373944 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.
%e A373944 Table starts:
%e A373944   1;
%e A373944   2,4,8,16;
%e A373944   3,6,9,12,18,24,27,32,...,288;
%e A373944   5,10,15,20,25,30,40,45,50,60,...,19200;
%e A373944   7,14,21,28,...,13829760;
%t A373944 (* First, load function f from A162306 *)
%t A373944 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 *)
%Y A373944 Cf. A000040, A002110, A002473, A003586, A007947, A051037, A051038, A080197, A080681, A080682, A080683.
%K A373944 nonn,easy
%O A373944 1,2
%A A373944 _David James Sycamore_, Jun 23 2024