A182944 -id:A182944 - cp's OEIS frontend

A357910 The natural numbers ordered lexicographically by their prime factorization, with prime factors written in decreasing order (see comments).

1, 2, 4, 3, 6, 8, 9, 12, 5, 10, 15, 30, 16, 27, 18, 25, 20, 45, 60, 7, 14, 21, 42, 35, 70, 105, 210, 32, 81, 24, 125, 40, 75, 90, 49, 28, 63, 84, 175, 140, 315, 420, 11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310, 64, 243, 36, 625, 50

Offset: 0

Michael De Vlieger, Jan 23 2023

Offset to match A019565, to which this sequence is analogous.
Let c(x) be the cardinality of x^e in a(1..n-1). Set multiplier m(x) = 1, increasing whereupon m(x)*x appears in the sequence.
Define function f(x) to be such that prime x implies x^c(x), otherwise x implies m(x)*x such that rad(m(x)) | rad(x).
Let b(n) be the expression of n through powers 2^e, thus, b(5) = "101" = 2^0 + 2^2. Then Heinz number h(n) = Product { prime(e+1) }, thus, h(n) = 10 = prime(0+1) * prime(2+1).
Define S(n) = { k : rad(k) = k, gpf(k) = prime(n) } such that the ordering of elements is according to h(k). Thus, S(4) = {7, 14, 21, 42, 35, 70, 105, 210} since, writing 1 when divisible by prime(j) in the j-th place read from the right, these are {1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111}, and interpreted as binary numbers these are {8, 9, 10, 11, 12, 13, 14, 15}. S(4) appears as row 4 of A019565 when interpreted as an irregular triangle.
We may generate T(n) by mapping f across S(1..n-1) and appending S(n), which is equivalent to mapping f across S(1..n). Note that we ignore the empty product, the sole element of S(0) = {1}, for n > 0.
Let R(k) = { r : rad(r) | rad(k) }. Hence, R(2) = A000079, R(6) = A003586, and R(10) = A003592. Let K = A019565(k). Column k contains K*R(K). Therefore, T(n,1) = A000079, T(n,3) = { 6*A003586 }, T(n,5) = { 10 * A003592 } and generally, for k = 2^e and p = prime(e+1), we have T(n, 2^e) = { p * p^m }.
R(k) is the tensor product of { p^e : e >= 0 } for all p | k and is therefore infinite for k > 1. A019565 is a permutation of squarefree numbers A005117, an infinite sequence. Therefore this sequence is infinite and a permutation of natural numbers.

			T(0) = S(0) = {1},
T(1) = S(1) = {2},
T(2) = {4, 3, 6}, since, mapping f across S(1..2) = {2, 3, 6};
  2 is prime and c(2) = 2, hence 2^c(2) = 4, but
  3 and 6 are new to this sequence, therefore {4, 3, 6}.
T(3) = {8, 9, 12, 5, 10, 15, 30}, since S(1..2) = {2, 3, 6};
  2 is prime and c(2) = 3, hence 2^c(2) = 2^3 = 8,
  3 is prime and c(3) = 2, hence 3^c(3) = 3^2 = 9,
  6 is composite and m(6) = 2, 2 | 6, hence 2*6 = 12,
  and we append S(3).
T(4) = {16, 27, 18, 25, 20, 45, 60, 7, 14, 21, 42, 35, 70, 105, 210},
  since S(1..3) = {2, 3, 6, 5, 10, 15, 30};
  2 is prime and c(2) = 4, hence 2^c(2) = 2^4 = 16,
  3 is prime and c(3) = 3, hence 3^c(3) = 3^3 = 27,
  6 is composite and m(6) = 3, 3 | 6, hence 3*6 = 18,
  5 is prime and c(5) = 2, hence 5^c(5) = 5^2 = 25,
  10 is composite and m(10) = 2, 2 | 10, hence 2*10 = 20,
  15 is composite and m(15) = 2, but 2 does not divide 15,
    therefore we increment m(15) = 3,
    3 | 15, hence 3*15 = 45.
  30 is composite and m(30) = 2, 2 | 30, hence 2*30 = 60,
  and we append S(3), etc. Let f(x) = phi(x)/x.
First terms of the first rows appear as follows:
  n\k    1      2    3       4    5     6    7       8
  ----------------------------------------------------
   1     2
   2     4      3    6
   3     8      9   12       5   10    15   30
   4    16     27   18      25   20    45   60       7  ...
   5    32     81   24     125   40    75   90      49  ...
   6    64    243   36     625   50   135  120     343  ...
   7   128    729   48    3125   80   225  150    2401  ...
   8   256   2187   54   15625  100   375  180   16807  ...
   9   512   6561   72   78125  160   405  240  117649  ...
  10  1024  19683   96  390625  200   675  270  823543  ...

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Log log scatterplot of a(n) n = 1..10^5.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..503, showing primes in red, composite prime powers in gold, squarefree composites in green, primorials in dark blue, and other numbers in light blue.
Michael De Vlieger, Plot prime(k)^e(k) | a(n) at (x,y) = (n,k) for n = 0..1024, showing multiplicity e(k) with a color function where black = 1, red = 2, ..., magenta = 9.

Cf. A000040, A000079, A002110, A003586, A003592, A007947, A019565, A182944.

nn = 16; rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; q[_] = 1; r = t[0] = {1}; c[1] = True; Do[Set[s, Join[r, Prime[n]*r]]; Set[t[n], Map[If[PrimeQ[#], Set[m, #^q[#]]; q[#]++; m, Set[m, q[#] #]; q[#]++; While[! Divisible[#, rad[q[#]]], q[#]++]; m] &, Rest[r]]]; r = s, {n, nn}]; {{1}}~Join~Rest@ Array[t, nn] // Flatten

cp's OEIS Frontend

A357910 The natural numbers ordered lexicographically by their prime factorization, with prime factors written in decreasing order (see comments).

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