A363250 Numbers in A363063 arranged in lexicographic order according to ordered prime signature (i.e., multiplicities of prime power factors p^k, written in order of p).

1, 2, 4, 12, 8, 24, 16, 48, 144, 720, 32, 96, 288, 1440, 864, 4320, 21600, 151200, 64, 192, 576, 2880, 1728, 8640, 43200, 302400, 128, 384, 1152, 5760, 3456, 17280, 86400, 604800, 10368, 51840, 259200, 1814400, 256, 768, 2304, 11520, 6912, 34560, 172800, 1209600, 20736, 103680, 518400, 3628800, 62208

Offset: 0

Author

Michael De Vlieger, May 23 2023

Keywords

• nonn
• tabf

Comments

The sequence is also readable as an irregular triangle by rows in which row n lists the terms divisible by 2^k but not by 2^(k+1).

Numbers m in A363063 are products of prime powers p(j)^S(j), j = 1..N, where p(j) is the j-th prime, such that p(j+1)^S(j+1) < p(j)^S(j). As consequence of definition of A363063, S(j) > S(j+1), hence multiplicities S(j) are distinct. Consequently, A363063 is a subset of A025487; m is a product of primorials. A025487 in turn is a subset of A055932.

These qualities enable us to write an algorithm that increments S(j) or drops the last term in S until we can increment S(j) to attain a solution. This algorithm generates terms in lexicographic order as described in the Name. The same qualities enable expression of m = Product p(j)^S(j) instead as Sum 2^(S(j)-1), a strictly increasing sequence.

Examples

			Table of n, a(n), and multiplicities S(j) written such that Product p(j)^S(j) = a(n). a(n) = A000079(i) is shown in the penultimate column, while a(n) = A347284(k) appears in the last column.
   n      a(n) multiplicities  i    k
  -----------------------------------
   0:       1                  0    0
   1:       2           1      1    1
   2:       4         2        2
   3:      12         2 1           2
   4:       8       3          3
   5:      24       3   1           3
   6:      16     4            4
   7:      48     4     1
   8:     144     4   2
   9:     720     4   2 1           4
  10:      32   5              5
  11:      96   5       1
  12:     288   5     2
  13:    1440   5     2 1
  14:     864   5   3
  15:    4320   5   3   1
  16:   21600   5   3 2
  17:  151200   5   3 2 1           5
  ...
Sequence read as an irregular triangle T(n, k):
  n\k   1    2    3     4     5     6      7       8
  ---------------------------------------------------
  0:    1
  1:    2
  2:    4   12
  3:    8   24
  4:   16   48  144   720
  5:   32   96  288  1440   864  4320  21600  151200
  6:   64  192  576  2880  1728  8640  43200  302400
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..14158 (rows i = 0..30, flattened)
• Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n,e), n = 1..3526, 12X vertical exaggeration

Crossrefs

Cf. A000079, A025487, A055932, A067255, A087980, A124010, A347284, A363063.

Subsequence of A362227.

Programs

Mathematica

nn = 12;
 f[x_] := Times @@ MapIndexed[Prime[First[#2]]^#1 &, x];
 {1}~Join~Reap[Do[s = {i}; Sow[2^i]; Set[k, 1];
     Do[
      If[Prime[k]^s[[-1]] > Prime[k + 1],
       AppendTo[s, 1]; k++; Sow[f[s]],
       If[Length[s] == 1, Break[],
        If[Prime[k - 1]^(s[[-2]]) > Prime[k]^(s[[-1]] + 1),
         s[[-1]]++; Sow[f[s]],
         While[And[k > 1,
           Prime[k - 1]^(s[[-2]]) < Prime[k]^(s[[-1]] + 1)], k--;
          s = s[[1 ;; k]]]; If[k == 1, Break[], s[[-1]]++; Sow[f[s]] ]
          ] ] ], {j, Infinity}], {i, nn}]][[-1, -1]]

Python

from sympy import nextprime,oo
from itertools import islice
primes = [2] # global list of first primes
def f(pi, ppmax):
    # Generate numbers with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
    if len(primes) <= pi: primes.append(nextprime(primes[-1]))
    p0 = primes[pi]
    if ppmax < p0:
        yield 1
        return
    pp = 1
    while pp <= ppmax:
        for x in f(pi+1, pp):
            yield pp*x
        pp *= p0
def A363250_list(nterms):
    return list(islice(f(0,oo),nterms)) # Pontus von Brömssen, May 25 2023

Formula

Seen as an irregular triangle, the first term in row i is 2^i, and the last term in row i is A347284(i).

Extensions

Edited by Michael De Vlieger/_Peter Munn_, May 27 2025