Factor-perfect numbers

Factor-perfect numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Consider the list of factors (divisors) of an integer, such as 12. The factors of 12 are [1, 2, 3, 4, 6, 12]. Consider all sorted sequences of the factors of n such that each succeeding number in such a sequence is a multiple of its predecessor. So, for 6, we have the factors (divisors) [1, 2, 3, 6]. The 3 unique lists of sequential multiples starting with 1 and ending with 6 that can be derived from these factors are [1, 6], [1, 2, 6], and [1, 3, 6].

Another way to see these sequences is as an set of all the ordered factorizations of a number taken so that their product is that number (excluding 1 from the sequence). So, for 6, we would have [6], [2, 3], and [3, 2]. In this description of the sequences, we are looking at the numbers needed to multiply by, in order to generate the next element in the sequences previously listed in our first definition of the sequence type, as we described it in the preceding paragraph, above.

For example, for the factorization of 6, if the first type of sequence is [1, 6], this is generated by [6] since 1 * 6 = 6. Similarly, the first type of sequence [1, 2, 6] is generated by the second type of sequence [2, 3] because 1 * 2 = 2 and 2 * 3 = 6. Similarly, [1, 3, 6] is generated by [3, 2] because 1 * 3 = 3 and 3 * 2 = 6.

If we count the number of such sorted sequences of multiples, or ordered factorizations, and using that count find all integers `n` for which the count of such sequences equals `n`, we have re-created the sequence of the "factor-perfect" numbers (OEIS 163272).

By some convention, on its OEIS page, the factor-perfect number sequence starts with 0 rather than 1. As might be expected with a sequence involving factorization and combinations, finding factor-perfect numbers becomes more demanding on CPU time as the numbers become large.

Task

• Show all 48 ordered sequences for each of the two methods for n = 48, which is the first non-trivial factor-perfect number.

According to the paper listed below by P. Erdos, the number of these sequences is

$$F(n)=\sum _{k}F({\frac {n}{a_{k}}})+1$$

where a is a list of the factors of n, including n, but excluding 1. F(n) is here the same as a function for calculating the number of different factorizations according to the second definition above except that F(1)=0 (where the number of factorizations of 1 must be 1 for it to be included in the sequence of factor-perfect numbers).

• Write a program to calculate and show the first 7 numbers of the factor-perfect numbers.

Stretch task

• Calculate and show more of the subsequent numbers in the sequence.

see also

OEIS A163272 (Numbers k such that k = A074206(k), the number of ordered factorizations of k)
OEIS A074206 (Kalmár's [Kalmar's] problem: number of ordered factorizations of n.)
On the maximal order of numbers in the “factorisatio numerorum” problem (Klazar/Luca)
On Some Asymptotic Formulas in The Theory of The "Factorisatio Numerorum" (P. Erdos)

J

Implementation:

factors=: {{/:~*/@>,{(^ i.)&.>/0 1+__ q:y}}
fp1=: {{ {{y#~0*/ .=~2|/\&>y}} y<@#"1~1,.~1,.#:i.2^_2+#y }}@factors
fp2=: 2 %~/\&.> fp1
Fi=: i.0
F=: {{
 if. y>:#Fi do. Fi=: Fi{.~1+y end.
 if. (1<y)*0=y{Fi do. Fi=: Fi y}~ 1++/F y%}.factors y end.
 y{Fi
}}"0

Task examples (formed into 8 columns for easy viewing):

   _8,\fp1 48
┌────────────┬───────────┬───────────┬──────────────┬──────────────┬───────────┬─────────────┬──────────────┐
│1 48        │1 24 48    │1 16 48    │1 12 48       │1 12 24 48    │1 8 48     │1 8 24 48    │1 8 16 48     │
├────────────┼───────────┼───────────┼──────────────┼──────────────┼───────────┼─────────────┼──────────────┤
│1 6 48      │1 6 24 48  │1 6 12 48  │1 6 12 24 48  │1 4 48        │1 4 24 48  │1 4 16 48    │1 4 12 48     │
├────────────┼───────────┼───────────┼──────────────┼──────────────┼───────────┼─────────────┼──────────────┤
│1 4 12 24 48│1 4 8 48   │1 4 8 24 48│1 4 8 16 48   │1 3 48        │1 3 24 48  │1 3 12 48    │1 3 12 24 48  │
├────────────┼───────────┼───────────┼──────────────┼──────────────┼───────────┼─────────────┼──────────────┤
│1 3 6 48    │1 3 6 24 48│1 3 6 12 48│1 3 6 12 24 48│1 2 48        │1 2 24 48  │1 2 16 48    │1 2 12 48     │
├────────────┼───────────┼───────────┼──────────────┼──────────────┼───────────┼─────────────┼──────────────┤
│1 2 12 24 48│1 2 8 48   │1 2 8 24 48│1 2 8 16 48   │1 2 6 48      │1 2 6 24 48│1 2 6 12 48  │1 2 6 12 24 48│
├────────────┼───────────┼───────────┼──────────────┼──────────────┼───────────┼─────────────┼──────────────┤
│1 2 4 48    │1 2 4 24 48│1 2 4 16 48│1 2 4 12 48   │1 2 4 12 24 48│1 2 4 8 48 │1 2 4 8 24 48│1 2 4 8 16 48 │
└────────────┴───────────┴───────────┴──────────────┴──────────────┴───────────┴─────────────┴──────────────┘
   _8,\fp2 48
┌───────┬───────┬───────┬─────────┬─────────┬───────┬─────────┬─────────┐
│48     │24 2   │16 3   │12 4     │12 2 2   │8 6    │8 3 2    │8 2 3    │
├───────┼───────┼───────┼─────────┼─────────┼───────┼─────────┼─────────┤
│6 8    │6 4 2  │6 2 4  │6 2 2 2  │4 12     │4 6 2  │4 4 3    │4 3 4    │
├───────┼───────┼───────┼─────────┼─────────┼───────┼─────────┼─────────┤
│4 3 2 2│4 2 6  │4 2 3 2│4 2 2 3  │3 16     │3 8 2  │3 4 4    │3 4 2 2  │
├───────┼───────┼───────┼─────────┼─────────┼───────┼─────────┼─────────┤
│3 2 8  │3 2 4 2│3 2 2 4│3 2 2 2 2│2 24     │2 12 2 │2 8 3    │2 6 4    │
├───────┼───────┼───────┼─────────┼─────────┼───────┼─────────┼─────────┤
│2 6 2 2│2 4 6  │2 4 3 2│2 4 2 3  │2 3 8    │2 3 4 2│2 3 2 4  │2 3 2 2 2│
├───────┼───────┼───────┼─────────┼─────────┼───────┼─────────┼─────────┤
│2 2 12 │2 2 6 2│2 2 4 3│2 2 3 4  │2 2 3 2 2│2 2 2 6│2 2 2 3 2│2 2 2 2 3│
└───────┴───────┴───────┴─────────┴─────────┴───────┴─────────┴─────────┘
   (#~ (=*>.F)) i.30000
0 1 48 1280 2496 28672 29808

jq

Adapted from Wren

Works with: jq

Also works with gojq, the Go implementation of jq provided a definition of _nwise is provided.

# unordered
def proper_divisors:
  . as $n
  | if $n > 1 then 1,
      ( range(2; 1 + (sqrt|floor)) as $i
        | if ($n % $i) == 0 then $i,
            (($n / $i) | if . == $i then empty else . end)
         else empty
	 end)
    else empty
    end;

# Uses the first definition and recursion to generate the sequences.
def moreMultiples($toSeq; $fromSeq):
    reduce $fromSeq[] as $i ({oneMores: []};
        if ($i > $toSeq[-1]) and ($i % $toSeq[-1]) == 0
	then .oneMores += [$toSeq + [$i]]
	else .
	end)
    | reduce range(0; .oneMores|length) as $i (.;
        .oneMores += moreMultiples(.oneMores[$i]; $fromSeq) )
    | .oneMores ;

# Input: {cache, ...}
# Output: {cache, count, ... }
def erdosFactorCount($n):
  def properDivisors: proper_divisors | select(. != 1);

  # Since this is a recursive function, the local and global states
  # must be managed separately:
  (reduce ($n|properDivisors) as $d ([0, .]; # count, global
        ($n/$d) as $t
	| ($t|tostring) as $ts
        | if .[1].cache|has($ts) then . else .[1].cache[$ts] = (.[1]|erdosFactorCount($t).count) end
        | .[0] += (.[1].cache[$ts])
    )) as $update
  | .count = $update[0] + 1
  | .cache = ($update[1].cache) ;

def task1:
  def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
  def neatly:  _nwise(4) | map(tostring|lpad(20)) | join(" ");
 
  moreMultiples([1]; [48|proper_divisors])
  | sort
  | map(. + [48]) + [[1, 48]]
  | "\(length) sequences using first definition:", neatly,

    (. as $listing
     | reduce range(0; $listing|length) as $i ([];
          $listing[$i] as $seq
          | (if ($seq[-1] != 48) then $seq + [48] else $seq end) as $seq
          | . + [[ range(1; $seq|length) as $i | ($seq[$i]/$seq[$i-1]) | floor ]] )
     | "\n\(length) sequences using second definition:", neatly );

# Stream the values of A163272:
def A163272:
  0,1,
  ({n:4}
  | while(true;
      .emit=null
      | erdosFactorCount(.n) # update the cache
      | if .count == .n then .emit =.n else . end
      | .n += 4 )
  | select(.emit).emit);

task1,
"",
"OEIS A163272:", limit(7; A163272)

Output:

48 sequences using first definition:
            [1,2,48]           [1,2,4,48]         [1,2,4,8,48]      [1,2,4,8,16,48]
     [1,2,4,8,24,48]        [1,2,4,12,48]     [1,2,4,12,24,48]        [1,2,4,16,48]
       [1,2,4,24,48]           [1,2,6,48]        [1,2,6,12,48]     [1,2,6,12,24,48]
       [1,2,6,24,48]           [1,2,8,48]        [1,2,8,16,48]        [1,2,8,24,48]
         [1,2,12,48]       [1,2,12,24,48]          [1,2,16,48]          [1,2,24,48]
            [1,3,48]           [1,3,6,48]        [1,3,6,12,48]     [1,3,6,12,24,48]
       [1,3,6,24,48]          [1,3,12,48]       [1,3,12,24,48]          [1,3,24,48]
            [1,4,48]           [1,4,8,48]        [1,4,8,16,48]        [1,4,8,24,48]
         [1,4,12,48]       [1,4,12,24,48]          [1,4,16,48]          [1,4,24,48]
            [1,6,48]          [1,6,12,48]       [1,6,12,24,48]          [1,6,24,48]
            [1,8,48]          [1,8,16,48]          [1,8,24,48]            [1,12,48]
        [1,12,24,48]            [1,16,48]            [1,24,48]               [1,48]

48 sequences using second definition:
              [2,24]             [2,2,12]            [2,2,2,6]          [2,2,2,2,3]
         [2,2,2,3,2]            [2,2,3,4]          [2,2,3,2,2]            [2,2,4,3]
           [2,2,6,2]              [2,3,8]            [2,3,2,4]          [2,3,2,2,2]
           [2,3,4,2]              [2,4,6]            [2,4,2,3]            [2,4,3,2]
             [2,6,4]            [2,6,2,2]              [2,8,3]             [2,12,2]
              [3,16]              [3,2,8]            [3,2,2,4]          [3,2,2,2,2]
           [3,2,4,2]              [3,4,4]            [3,4,2,2]              [3,8,2]
              [4,12]              [4,2,6]            [4,2,2,3]            [4,2,3,2]
             [4,3,4]            [4,3,2,2]              [4,4,3]              [4,6,2]
               [6,8]              [6,2,4]            [6,2,2,2]              [6,4,2]
               [8,6]              [8,2,3]              [8,3,2]               [12,4]
            [12,2,2]               [16,3]               [24,2]                 [48]

OEIS A163272:
0
1
48
1280
2496
28672
29808

Julia

Revised to reflect a faster counting method (see second paper in the references).

using Primes
using Memoize

""" Return the factors of n, including 1, n """
function factors(n::T)::Vector{T} where T <: Integer
  sort(vec(map(prod, Iterators.product((p.^(0:m) for (p, m) in eachfactor(n))...))))
end

""" Uses the first definition and recursion to generate the sequences """
function more_multiples(to_seq, from_seq)
    onemores = [[to_seq; i] for i in from_seq if i > to_seq[end] && i % to_seq[end] == 0]
    isempty(onemores) && return Int[]
    return append!(onemores, mapreduce(seq -> more_multiples(seq, from_seq), append!, onemores))
end


""" See reference paper by Erdos, page 1 """
@memoize function kfactors(n)
    a = factors(n)
    return sum(kfactors(n ÷ d) for d in a[begin+1:end]) + 1
end

listing = sort!(push!(map(a -> push!(a, 48), more_multiples([1], factors(48)[begin+1:end-1])), [1, 48]))
println("48 sequences using first definition:")
for (i, seq) in enumerate(listing)
    print(rpad(seq, 22), i % 4 == 0 ? "\n" : "")
end

println("\n48 sequences using second definition:")
for (i, seq) in enumerate(listing)
    seq2 = [seq[j] ÷ seq[j - 1] for j in 2:length(seq)]
    print(rpad(seq2, 20), i % 4 == 0 ? "\n" : "")
end

println("\nOEIS A163272: ")
for n in 0:2_400_000
    if n == 0 || kfactors(n) == n
        print(n, ",  ")
    end
end

Output:

48 sequences using first definition:
[1, 2, 4, 8, 16, 48]  [1, 2, 4, 8, 24, 48]  [1, 2, 4, 8, 48]      [1, 2, 4, 12, 24, 48]
[1, 2, 4, 12, 48]     [1, 2, 4, 16, 48]     [1, 2, 4, 24, 48]     [1, 2, 4, 48]
[1, 2, 6, 12, 24, 48] [1, 2, 6, 12, 48]     [1, 2, 6, 24, 48]     [1, 2, 6, 48]
[1, 2, 8, 16, 48]     [1, 2, 8, 24, 48]     [1, 2, 8, 48]         [1, 2, 12, 24, 48]
[1, 2, 12, 48]        [1, 2, 16, 48]        [1, 2, 24, 48]        [1, 2, 48]
[1, 3, 6, 12, 24, 48] [1, 3, 6, 12, 48]     [1, 3, 6, 24, 48]     [1, 3, 6, 48]
[1, 3, 12, 24, 48]    [1, 3, 12, 48]        [1, 3, 24, 48]        [1, 3, 48]
[1, 4, 8, 16, 48]     [1, 4, 8, 24, 48]     [1, 4, 8, 48]         [1, 4, 12, 24, 48]
[1, 4, 12, 48]        [1, 4, 16, 48]        [1, 4, 24, 48]        [1, 4, 48]
[1, 6, 12, 24, 48]    [1, 6, 12, 48]        [1, 6, 24, 48]        [1, 6, 48]
[1, 8, 16, 48]        [1, 8, 24, 48]        [1, 8, 48]            [1, 12, 24, 48]
[1, 12, 48]           [1, 16, 48]           [1, 24, 48]           [1, 48]

48 sequences using second definition:
[2, 2, 2, 2, 3]     [2, 2, 2, 3, 2]     [2, 2, 2, 6]        [2, 2, 3, 2, 2]
[2, 2, 3, 4]        [2, 2, 4, 3]        [2, 2, 6, 2]        [2, 2, 12]
[2, 3, 2, 2, 2]     [2, 3, 2, 4]        [2, 3, 4, 2]        [2, 3, 8]
[2, 4, 2, 3]        [2, 4, 3, 2]        [2, 4, 6]           [2, 6, 2, 2]
[2, 6, 4]           [2, 8, 3]           [2, 12, 2]          [2, 24]
[3, 2, 2, 2, 2]     [3, 2, 2, 4]        [3, 2, 4, 2]        [3, 2, 8]
[3, 4, 2, 2]        [3, 4, 4]           [3, 8, 2]           [3, 16]
[4, 2, 2, 3]        [4, 2, 3, 2]        [4, 2, 6]           [4, 3, 2, 2]
[4, 3, 4]           [4, 4, 3]           [4, 6, 2]           [4, 12]
[6, 2, 2, 2]        [6, 2, 4]           [6, 4, 2]           [6, 8]
[8, 2, 3]           [8, 3, 2]           [8, 6]              [12, 2, 2]
[12, 4]             [16, 3]             [24, 2]             [48]

OEIS A163272:
0,  1,  48,  1280,  2496,  28672,  29808,  454656,  2342912,  

Nim

Translation of: Python

import std/[algorithm, strutils, sugar, tables]

func moreMultiples(toSeq, fromSeq: seq[int]): seq[seq[int]] =
  ## Uses the first definition and recursion to generate the sequences.
  result = collect:
             for i in fromSeq:
               if i > toSeq[^1] and i mod toSeq[^1] == 0:
                 toSeq & i

  for i in 0..result.high:
    for arr in moreMultiples(result[i], fromSeq):
      result.add arr

func divisors(n: int): seq[int] =
  ## Return the list of divisors of "n".
  var d = 1
  while d * d <= n:
    if n mod d == 0:
      let q = n div d
      result.add d
      if q != d:
        result.add q
    inc d
  result.sort()

func cmp(x, y: seq[int]): int =
  ## Compare two sequences.
  for i in 0..<min(x.len, y.len):
    result = cmp(x[i], y[i])
    if result != 0: return
  result = cmp(x.len, y.len)

let listing = collect(
                for a in sorted(moreMultiples(@[1], divisors(48)[1..^2]), cmp):
                  a & 48) & @[@[1, 48]]

echo "48 sequences using first definition:"
for i, s in listing:
  let item = '[' & s.join(", ") & ']'
  stdout.write alignLeft(item, 22)
  stdout.write if i mod 4 == 3: '\n' else: ' '

# Derive second definition's sequences
echo "\n48 sequences using second definition:"

for i, s1 in listing:
  let s2 = collect:
             for j in 1..s1.high:
               s1[j] div s1[j - 1]
  let item = '[' & s2.join(", ") & ']'
  stdout.write alignLeft(item, 20)
  stdout.write if i mod 4 == 3: '\n' else: ' '

var cache: Table[int, int]

proc erdosFactorCount(n: int): int =
  ## Erdos method.
  if n in cache: return cache[n]
  let ds = divisors(n)
  if ds.len >= 2:
    for d in ds[1..^2]:
      result += erdosFactorCount(n div d)
  inc result
  cache[n] = result

stdout.write "\nOEIS A163272: "
let s = collect:
          for num in 0..<2_400_000:
            if num == 0 or erdosFactorCount(num) == num:
              num
echo s.join(", ")

Output:

48 sequences using first definition:
[1, 2, 48]             [1, 2, 4, 48]          [1, 2, 4, 8, 48]       [1, 2, 4, 8, 16, 48]  
[1, 2, 4, 8, 24, 48]   [1, 2, 4, 12, 48]      [1, 2, 4, 12, 24, 48]  [1, 2, 4, 16, 48]     
[1, 2, 4, 24, 48]      [1, 2, 6, 48]          [1, 2, 6, 12, 48]      [1, 2, 6, 12, 24, 48] 
[1, 2, 6, 24, 48]      [1, 2, 8, 48]          [1, 2, 8, 16, 48]      [1, 2, 8, 24, 48]     
[1, 2, 12, 48]         [1, 2, 12, 24, 48]     [1, 2, 16, 48]         [1, 2, 24, 48]        
[1, 3, 48]             [1, 3, 6, 48]          [1, 3, 6, 12, 48]      [1, 3, 6, 12, 24, 48] 
[1, 3, 6, 24, 48]      [1, 3, 12, 48]         [1, 3, 12, 24, 48]     [1, 3, 24, 48]        
[1, 4, 48]             [1, 4, 8, 48]          [1, 4, 8, 16, 48]      [1, 4, 8, 24, 48]     
[1, 4, 12, 48]         [1, 4, 12, 24, 48]     [1, 4, 16, 48]         [1, 4, 24, 48]        
[1, 6, 48]             [1, 6, 12, 48]         [1, 6, 12, 24, 48]     [1, 6, 24, 48]        
[1, 8, 48]             [1, 8, 16, 48]         [1, 8, 24, 48]         [1, 12, 48]           
[1, 12, 24, 48]        [1, 16, 48]            [1, 24, 48]            [1, 48]               

48 sequences using second definition:
[2, 24]              [2, 2, 12]           [2, 2, 2, 6]         [2, 2, 2, 2, 3]     
[2, 2, 2, 3, 2]      [2, 2, 3, 4]         [2, 2, 3, 2, 2]      [2, 2, 4, 3]        
[2, 2, 6, 2]         [2, 3, 8]            [2, 3, 2, 4]         [2, 3, 2, 2, 2]     
[2, 3, 4, 2]         [2, 4, 6]            [2, 4, 2, 3]         [2, 4, 3, 2]        
[2, 6, 4]            [2, 6, 2, 2]         [2, 8, 3]            [2, 12, 2]          
[3, 16]              [3, 2, 8]            [3, 2, 2, 4]         [3, 2, 2, 2, 2]     
[3, 2, 4, 2]         [3, 4, 4]            [3, 4, 2, 2]         [3, 8, 2]           
[4, 12]              [4, 2, 6]            [4, 2, 2, 3]         [4, 2, 3, 2]        
[4, 3, 4]            [4, 3, 2, 2]         [4, 4, 3]            [4, 6, 2]           
[6, 8]               [6, 2, 4]            [6, 2, 2, 2]         [6, 4, 2]           
[8, 6]               [8, 2, 3]            [8, 3, 2]            [12, 4]             
[12, 2, 2]           [16, 3]              [24, 2]              [48]                

OEIS A163272: 0, 1, 48, 1280, 2496, 28672, 29808, 454656, 2342912

Perl

Translation of: Raku

use v5.36;

sub table (@V) { my $t = 3 * (my $w = 2 + 20); ( sprintf( ('%-'.$w.'s')x@V, @V) ) =~ s/.{1,$t}\K/\n/gr }

sub proper_divisors ($x) {
    my @l;
    @l = 1 if $x > 1;
    for my $d (2 .. int sqrt $x) {
        if (0 == $x % $d) { push @l, $d; my $y = int($x/$d); push @l, $y if $y != $d }
    }
    @l
}

sub erdosFactorCount ($n) {
    my @foo = proper_divisors($n); shift @foo;
    state %cache;
    my ($sum,@divs) = (0, @foo); #(proper_divisors $n)[1..*]);
    for my $d (@divs) {
        my $t = int($n/$d);
        $cache{$t} = erdosFactorCount($t) unless $cache{$t};
        $sum += $cache{$t}
    }
    ++$sum
}

sub moreMultiples ($to, $from) {
    my @oneMores;
    for my $j (@$from) {
        push @oneMores, [@$to, $j] if $j > $$to[-1] && 0 == $j % $$to[-1]
    }
    return unless @oneMores;
    for (0 .. $#oneMores) {
        push @oneMores, moreMultiples($oneMores[$_], $from);
    }
    @oneMores
}

my @listing = [1];
push @listing, moreMultiples [1], [proper_divisors(48)];
map { push @$_, 48 } @listing;

my @lists; map { push @lists, join ' ', @$_ } @listing;
say @listing . " sequences using first definition:\n" . table(@lists);

my @listing2;
for my $j (0.. $#listing) {
    my @seq = @{$listing[$j]};
    push @seq, 48 if $seq[-1] != 48;
    push @listing2, join ' ', map { int $seq[$_] / $seq[$_-1] } 1 .. $#seq;
}

say @listing2 . " sequences using second definition:\n" . table(@listing2);

my($n,@fpns) = (4, 0,1);
while ($#fpns < 6) { push(@fpns, $n) if erdosFactorCount($n) == $n; $n += 4 }
say "OEIS A163272: @fpns";

Output:

48 sequences using first definition:
1 48                  1 2 48                1 24 48
1 3 48                1 16 48               1 4 48
1 12 48               1 6 48                1 8 48
1 2 24 48             1 2 16 48             1 2 4 48
1 2 12 48             1 2 6 48              1 2 8 48
1 2 4 24 48           1 2 4 16 48           1 2 4 12 48
1 2 4 8 48            1 2 4 12 24 48        1 2 4 8 24 48
1 2 4 8 16 48         1 2 12 24 48          1 2 6 24 48
1 2 6 12 48           1 2 6 12 24 48        1 2 8 24 48
1 2 8 16 48           1 3 24 48             1 3 12 48
1 3 6 48              1 3 12 24 48          1 3 6 24 48
1 3 6 12 48           1 3 6 12 24 48        1 4 24 48
1 4 16 48             1 4 12 48             1 4 8 48
1 4 12 24 48          1 4 8 24 48           1 4 8 16 48
1 12 24 48            1 6 24 48             1 6 12 48
1 6 12 24 48          1 8 24 48             1 8 16 48

48 sequences using second definition:
48                    2 24                  24 2
3 16                  16 3                  4 12
12 4                  6 8                   8 6
2 12 2                2 8 3                 2 2 12
2 6 4                 2 3 8                 2 4 6
2 2 6 2               2 2 4 3               2 2 3 4
2 2 2 6               2 2 3 2 2             2 2 2 3 2
2 2 2 2 3             2 6 2 2               2 3 4 2
2 3 2 4               2 3 2 2 2             2 4 3 2
2 4 2 3               3 8 2                 3 4 4
3 2 8                 3 4 2 2               3 2 4 2
3 2 2 4               3 2 2 2 2             4 6 2
4 4 3                 4 3 4                 4 2 6
4 3 2 2               4 2 3 2               4 2 2 3
12 2 2                6 4 2                 6 2 4
6 2 2 2               8 3 2                 8 2 3

OEIS A163272: 0 1 48 1280 2496 28672 29808

Phix

Library: Phix/online

You can run this online here (expect a blank screen for ~30s).

--
-- demo/rosetta/factor-perfect_numbers.exw
--
with javascript_semantics

function get_factor_set(integer x)
    if x=1 then return {1} end if
    sequence res = {}
    for k=1 to x-1 do
        if remainder(x,k)=0 then
            for y in get_factor_set(k) do
                res = append(res,y&x)
            end for
        end if
    end for
    res = sort(res)
    return res
end function

function m(sequence s, integer f)
    sequence res = {}   
    for x in s do
        x = deep_copy(x)
        if x[$]!=f then x &= f end if
        for i=length(x) to 2 by -1 do
            x[i] /= x[i-1]
        end for
        res = append(res,x[2..$])
    end for
    return res
end function

constant N = 48
sequence rN = get_factor_set(N)
function jbm(bool munge)
    if munge then rN = m(rN,N) end if
    return {length(rN),join_by(apply(rN,ppf),1,4," ",fmt:="%-16s")}
end function
ppOpt({pp_IntCh,false,pp_StrFmt,3})
printf(1,"%d sequences using first definition:\n%s\n",jbm(false))
printf(1,"%d sequences using second definition:\n%s\n",jbm(true))

integer efc_cache = new_dict()

function erdosFactorCount(integer n)
    sequence divs = factors(n)
    integer res = 1
    for d in divs do
        integer t = n/d, r, node = getd_index(t,efc_cache)
        if node=NULL then
            r = erdosFactorCount(t)
            setd(t,r,efc_cache)
        else
            r = getd_by_index(node,efc_cache)
        end if
        res += r
    end for
    return res
end function

atom t = time(), t1 = t+1
integer n = 4
sequence res = {"0","1"}
while length(res)<iff(platform()=JS?8:9) do
    if erdosFactorCount(n)=n then
        res = append(res,sprintf("%d",n))
    end if
    n += 4
    if time()>t1 then
        progress("%d found, checking %d...\r",{length(res),n})
        t1 = time()+1
    end if
end while
progress("")
printf(1,"Found %d: %s (%s)\n",{length(res),join(res," "),elapsed(time()-t)})

wait_key()

Output:

48 sequences using first definition:
{1,2,4,8,16,48}  {1,2,4,8,24,48}  {1,2,4,8,48}     {1,2,4,12,24,48}
{1,2,4,12,48}    {1,2,4,16,48}    {1,2,4,24,48}    {1,2,4,48}
{1,2,6,12,24,48} {1,2,6,12,48}    {1,2,6,24,48}    {1,2,6,48}
{1,2,8,16,48}    {1,2,8,24,48}    {1,2,8,48}       {1,2,12,24,48}
{1,2,12,48}      {1,2,16,48}      {1,2,24,48}      {1,2,48}
{1,3,6,12,24,48} {1,3,6,12,48}    {1,3,6,24,48}    {1,3,6,48}
{1,3,12,24,48}   {1,3,12,48}      {1,3,24,48}      {1,3,48}
{1,4,8,16,48}    {1,4,8,24,48}    {1,4,8,48}       {1,4,12,24,48}
{1,4,12,48}      {1,4,16,48}      {1,4,24,48}      {1,4,48}
{1,6,12,24,48}   {1,6,12,48}      {1,6,24,48}      {1,6,48}
{1,8,16,48}      {1,8,24,48}      {1,8,48}         {1,12,24,48}
{1,12,48}        {1,16,48}        {1,24,48}        {1,48}

48 sequences using second definition:
{2,2,2,2,3}      {2,2,2,3,2}      {2,2,2,6}        {2,2,3,2,2}
{2,2,3,4}        {2,2,4,3}        {2,2,6,2}        {2,2,12}
{2,3,2,2,2}      {2,3,2,4}        {2,3,4,2}        {2,3,8}
{2,4,2,3}        {2,4,3,2}        {2,4,6}          {2,6,2,2}
{2,6,4}          {2,8,3}          {2,12,2}         {2,24}
{3,2,2,2,2}      {3,2,2,4}        {3,2,4,2}        {3,2,8}
{3,4,2,2}        {3,4,4}          {3,8,2}          {3,16}
{4,2,2,3}        {4,2,3,2}        {4,2,6}          {4,3,2,2}
{4,3,4}          {4,4,3}          {4,6,2}          {4,12}
{6,2,2,2}        {6,2,4}          {6,4,2}          {6,8}
{8,2,3}          {8,3,2}          {8,6}            {12,2,2}
{12,4}           {16,3}           {24,2}           {48}

Found 9: 0 1 48 1280 2496 28672 29808 454656 2342912 (1 minute and 9s)

Unfortunately it takes 4 minutes 13 seconds to find 9 under p2js, so I've limited that to 8 (as mentioned above, ~30s)

Python

''' Rosetta Code task Factor-perfect_numbers '''

from functools import cache
from sympy import divisors


def more_multiples(to_seq, from_seq):
    ''' Uses the first definition and recursion to generate the sequences '''
    onemores = [to_seq + [i]
                for i in from_seq if i > to_seq[-1] and i % to_seq[-1] == 0]
    if len(onemores) == 0:
        return []
    for i in range(len(onemores)):
        for arr in more_multiples(onemores[i], from_seq):
            onemores.append(arr)
    return onemores


listing = [a + [48]
           for a in sorted(more_multiples([1], divisors(48)[1:-1]))] + [[1, 48]]
print('48 sequences using first definition:')
for j, seq in enumerate(listing):
    print(f'{str(seq):22}', end='\n' if (j + 1) % 4 == 0 else '')


# Derive second definition's sequences
print('\n48 sequences using second definition:')
for k, seq in enumerate(listing):
    seq2 = [seq[i] // seq[i - 1] for i in range(1, len(seq))]
    print(f'{str(seq2):20}', end='\n' if (k + 1) % 4 == 0 else '')


@cache
def erdos_factor_count(number):
    ''' 'Erdos method '''
    return sum(erdos_factor_count(number // d) for d in divisors(number)[1:-1]) + 1


print("\nOEIS A163272:  ", end='')
for num in range(2_400_000):
    if num == 0 or erdos_factor_count(num) == num:
        print(num, end=',  ')

Output:

48 sequences using first definition:
[1, 2, 48]            [1, 2, 4, 48]         [1, 2, 4, 8, 48]      [1, 2, 4, 8, 16, 48]  
[1, 2, 4, 8, 24, 48]  [1, 2, 4, 12, 48]     [1, 2, 4, 12, 24, 48] [1, 2, 4, 16, 48]     
[1, 2, 4, 24, 48]     [1, 2, 6, 48]         [1, 2, 6, 12, 48]     [1, 2, 6, 12, 24, 48] 
[1, 2, 6, 24, 48]     [1, 2, 8, 48]         [1, 2, 8, 16, 48]     [1, 2, 8, 24, 48]     
[1, 2, 12, 48]        [1, 2, 12, 24, 48]    [1, 2, 16, 48]        [1, 2, 24, 48]        
[1, 3, 48]            [1, 3, 6, 48]         [1, 3, 6, 12, 48]     [1, 3, 6, 12, 24, 48] 
[1, 3, 6, 24, 48]     [1, 3, 12, 48]        [1, 3, 12, 24, 48]    [1, 3, 24, 48]        
[1, 4, 48]            [1, 4, 8, 48]         [1, 4, 8, 16, 48]     [1, 4, 8, 24, 48]     
[1, 4, 12, 48]        [1, 4, 12, 24, 48]    [1, 4, 16, 48]        [1, 4, 24, 48]        
[1, 6, 48]            [1, 6, 12, 48]        [1, 6, 12, 24, 48]    [1, 6, 24, 48]        
[1, 8, 48]            [1, 8, 16, 48]        [1, 8, 24, 48]        [1, 12, 48]           
[1, 12, 24, 48]       [1, 16, 48]           [1, 24, 48]           [1, 48]                            

48 sequences using second definition:
[2, 24]             [2, 2, 12]          [2, 2, 2, 6]        [2, 2, 2, 2, 3]     
[2, 2, 2, 3, 2]     [2, 2, 3, 4]        [2, 2, 3, 2, 2]     [2, 2, 4, 3]        
[2, 2, 6, 2]        [2, 3, 8]           [2, 3, 2, 4]        [2, 3, 2, 2, 2]     
[2, 3, 4, 2]        [2, 4, 6]           [2, 4, 2, 3]        [2, 4, 3, 2]        
[2, 6, 4]           [2, 6, 2, 2]        [2, 8, 3]           [2, 12, 2]          
[3, 16]             [3, 2, 8]           [3, 2, 2, 4]        [3, 2, 2, 2, 2]     
[3, 2, 4, 2]        [3, 4, 4]           [3, 4, 2, 2]        [3, 8, 2]           
[4, 12]             [4, 2, 6]           [4, 2, 2, 3]        [4, 2, 3, 2]        
[4, 3, 4]           [4, 3, 2, 2]        [4, 4, 3]           [4, 6, 2]           
[6, 8]              [6, 2, 4]           [6, 2, 2, 2]        [6, 4, 2]           
[8, 6]              [8, 2, 3]           [8, 3, 2]           [12, 4]             
[12, 2, 2]          [16, 3]             [24, 2]             [48]                

OEIS A163272:  0,  1,  48,  1280,  2496,  28672,  29808,  454656,  2342912,  

Raku

Translation of: Wren

# 20221029 Raku programming solution

sub propdiv (\x) {
   my @l = 1 if x > 1;
   for (2 .. x.sqrt.floor) -> \d {
      unless x % d { @l.push: d; my \y = x div d; @l.push: y if y != d }
   }
   @l
}

sub moreMultiples (@toSeq, @fromSeq) {
   my @oneMores = gather for @fromSeq -> \j {
      take @toSeq.clone.push(j) if j > @toSeq[*-1] and j %% @toSeq[*-1]
   }
   return () unless @oneMores.Bool;
   for 0..^@oneMores {
      @oneMores.append: moreMultiples @oneMores[$_], @fromSeq
   }
   @oneMores
}

sub erdosFactorCount (\n) {
   state %cache;
   my ($sum,@divs) = 0, |(propdiv n)[1..*];
   for @divs -> \d {
      unless %cache{my \t = n div d}:exists { %cache{t} = erdosFactorCount(t) }
      $sum += %cache{t}
   }
   ++$sum
}

my @listing = moreMultiples [1], propdiv(48);
given @listing { $_.map: *.push: 48; $_.push: [1,48] }
say @listing.elems," sequences using first definition:";
for @listing.rotor(4) -> \line { line.map: { printf "%-20s", $_ } ; say() }

my @listing2 = gather for (0..^+@listing) -> \j {
   my @seq = |@listing[j];
   @seq.append: 48 if @seq[*-1] != 48;
   take (1..^@seq).map: { @seq[$_] div @seq[$_-1] }
}
say "\n{@listing2.elems} sequences using second definition:";
for @listing2.rotor(4) -> \line { line.map: { printf "%-20s", $_ } ; say() }

say "\nOEIS A163272:";
my ($n,@fpns) = 4, 0,1;
while (@fpns < 7) { @fpns.push($n) if erdosFactorCount($n) == $n; $n += 4 }
say ~@fpns;

Output:

48 sequences using first definition:
1 2 48              1 24 48             1 3 48              1 16 48             
1 4 48              1 12 48             1 6 48              1 8 48              
1 2 24 48           1 2 16 48           1 2 4 48            1 2 12 48           
1 2 6 48            1 2 8 48            1 2 4 24 48         1 2 4 16 48         
1 2 4 12 48         1 2 4 8 48          1 2 4 12 24 48      1 2 4 8 24 48       
1 2 4 8 16 48       1 2 12 24 48        1 2 6 24 48         1 2 6 12 48         
1 2 6 12 24 48      1 2 8 24 48         1 2 8 16 48         1 3 24 48           
1 3 12 48           1 3 6 48            1 3 12 24 48        1 3 6 24 48         
1 3 6 12 48         1 3 6 12 24 48      1 4 24 48           1 4 16 48           
1 4 12 48           1 4 8 48            1 4 12 24 48        1 4 8 24 48         
1 4 8 16 48         1 12 24 48          1 6 24 48           1 6 12 48           
1 6 12 24 48        1 8 24 48           1 8 16 48           1 48                

48 sequences using second definition:
2 24                24 2                3 16                16 3                
4 12                12 4                6 8                 8 6                 
2 12 2              2 8 3               2 2 12              2 6 4               
2 3 8               2 4 6               2 2 6 2             2 2 4 3             
2 2 3 4             2 2 2 6             2 2 3 2 2           2 2 2 3 2           
2 2 2 2 3           2 6 2 2             2 3 4 2             2 3 2 4             
2 3 2 2 2           2 4 3 2             2 4 2 3             3 8 2               
3 4 4               3 2 8               3 4 2 2             3 2 4 2             
3 2 2 4             3 2 2 2 2           4 6 2               4 4 3               
4 3 4               4 2 6               4 3 2 2             4 2 3 2             
4 2 2 3             12 2 2              6 4 2               6 2 4               
6 2 2 2             8 3 2               8 2 3               48                  

OEIS A163272:
0 1 48 1280 2496 28672 29808

Sidef

Translation of: Perl

func erdosFactorCount (n) is cached {

    var sum = 1
    var divs = proper_divisors(n).slice(1)

    divs.each {|d|
        sum += __FUNC__(idiv(n,d))
    }

    return sum
}

func moreMultiples (to, from) {
    var oneMores = []
    from.each {|j|
        if (j > to.tail && to.tail.divides(j)) {
            oneMores << [to..., j]
        }
    }
    for k in (oneMores.range) {
        oneMores << __FUNC__(oneMores[k], from)...
    }
    return oneMores
}

var listing = [[1]]
listing << moreMultiples([1], proper_divisors(48))...
listing.each {|a| a << 48 }

say "#{listing.len} sequences using first definition:"
listing.slices(3).each { .map { .join(' ') }.map{ '%-20s' % _ }.join.say }

var listing2 = gather {
    for j in (^listing.len) {
        var seq = listing[j]
        take(1..seq.end -> map {|j| seq[j] / seq[j-1] })
    }
}

say "\n#{listing2.len} sequences using second definition:"
listing2.slices(3).each { .map { .join(' ') }.map{ '%-20s' % _ }.join.say }

print "\nOEIS A163272: "
say [0, 1, (1..Inf -> lazy.map {|n| 4*n }.grep{|n| erdosFactorCount(n) == n }.first(5))...]

Output:

48 sequences using first definition:
1 48                1 2 48              1 3 48              
1 4 48              1 6 48              1 8 48              
1 12 48             1 16 48             1 24 48             
1 2 4 48            1 2 6 48            1 2 8 48            
1 2 12 48           1 2 16 48           1 2 24 48           
1 2 4 8 48          1 2 4 12 48         1 2 4 16 48         
1 2 4 24 48         1 2 4 8 16 48       1 2 4 8 24 48       
1 2 4 12 24 48      1 2 6 12 48         1 2 6 24 48         
1 2 6 12 24 48      1 2 8 16 48         1 2 8 24 48         
1 2 12 24 48        1 3 6 48            1 3 12 48           
1 3 24 48           1 3 6 12 48         1 3 6 24 48         
1 3 6 12 24 48      1 3 12 24 48        1 4 8 48            
1 4 12 48           1 4 16 48           1 4 24 48           
1 4 8 16 48         1 4 8 24 48         1 4 12 24 48        
1 6 12 48           1 6 24 48           1 6 12 24 48        
1 8 16 48           1 8 24 48           1 12 24 48          

48 sequences using second definition:
48                  2 24                3 16                
4 12                6 8                 8 6                 
12 4                16 3                24 2                
2 2 12              2 3 8               2 4 6               
2 6 4               2 8 3               2 12 2              
2 2 2 6             2 2 3 4             2 2 4 3             
2 2 6 2             2 2 2 2 3           2 2 2 3 2           
2 2 3 2 2           2 3 2 4             2 3 4 2             
2 3 2 2 2           2 4 2 3             2 4 3 2             
2 6 2 2             3 2 8               3 4 4               
3 8 2               3 2 2 4             3 2 4 2             
3 2 2 2 2           3 4 2 2             4 2 6               
4 3 4               4 4 3               4 6 2               
4 2 2 3             4 2 3 2             4 3 2 2             
6 2 4               6 4 2               6 2 2 2             
8 2 3               8 3 2               12 2 2              

OEIS A163272: [0, 1, 48, 1280, 2496, 28672, 29808]

Wren

Translation of: Python

Library: Wren-math

Library: Wren-fmt

Timings are about: 0.19 secs for 7, 8.5 secs for 8 and 97 secs for 9 factor-perfect numbers.

import "./math" for Int, Nums
import "./fmt" for Fmt

// Uses the first definition and recursion to generate the sequences.
var moreMultiples
moreMultiples = Fn.new { |toSeq, fromSeq|
    var oneMores = []
    for (i in fromSeq) {
        if (i > toSeq[-1] && i%toSeq[-1] == 0) oneMores.add(toSeq + [i])
    }
    if (oneMores.isEmpty) return []
    for (i in 0...oneMores.count) {
        oneMores.addAll(moreMultiples.call(oneMores[i], fromSeq))
    }
    return oneMores
}

var cache = {}

var erdosFactorCount
erdosFactorCount = Fn.new { |n|
    var divs = Int.properDivisors(n)
    divs.removeAt(0)
    var sum = 0
    for (d in divs) {
        var t = (n/d).floor
        if (!cache.containsKey(t)) cache[t] = erdosFactorCount.call(t)
        sum = sum + cache[t]
    }
    return sum + 1
}

var listing = moreMultiples.call([1], Int.properDivisors(48))
listing.sort { |l1, l2|
    var c1 = l1.count
    var c2 = l2.count
    for (i in 1...c1.min(c2)) {
        if (l1[i] < l2[i]) return true
        if (l1[i] > l2[i]) return false
    }
    if (c1 < c2) return true
    return false
}
listing.each { |l| l.add(48) }
listing.add([1, 48])
System.print("%(listing.count) sequences using first definition:")
Fmt.tprint("$-21n", listing, 4)

System.print("\n%(listing.count) sequences using second definition:")
var listing2 = []
for (i in 0...listing.count) {
    var seq = listing[i]
    if (seq[-1] != 48) seq.add(48)
    var seq2 = (1...seq.count).map { |i| (seq[i]/seq[i-1]).floor }.toList
    listing2.add(seq2)
}
Fmt.tprint("$-17n", listing2, 4)

System.print("\nOEIS A163272:")
var n = 4
var fpns = [0, 1]
while (fpns.count < 9) {
    if (erdosFactorCount.call(n) == n) fpns.add(n)
    n = n + 4
}
System.print(fpns)

Output:

48 sequences using first definition:
[1, 2, 48]            [1, 2, 4, 48]         [1, 2, 4, 8, 48]      [1, 2, 4, 8, 16, 48]  
[1, 2, 4, 8, 24, 48]  [1, 2, 4, 12, 48]     [1, 2, 4, 12, 24, 48] [1, 2, 4, 16, 48]     
[1, 2, 4, 24, 48]     [1, 2, 6, 48]         [1, 2, 6, 12, 48]     [1, 2, 6, 12, 24, 48] 
[1, 2, 6, 24, 48]     [1, 2, 8, 48]         [1, 2, 8, 16, 48]     [1, 2, 8, 24, 48]     
[1, 2, 12, 48]        [1, 2, 12, 24, 48]    [1, 2, 16, 48]        [1, 2, 24, 48]        
[1, 3, 48]            [1, 3, 6, 48]         [1, 3, 6, 12, 48]     [1, 3, 6, 12, 24, 48] 
[1, 3, 6, 24, 48]     [1, 3, 12, 48]        [1, 3, 12, 24, 48]    [1, 3, 24, 48]        
[1, 4, 48]            [1, 4, 8, 48]         [1, 4, 8, 16, 48]     [1, 4, 8, 24, 48]     
[1, 4, 12, 48]        [1, 4, 12, 24, 48]    [1, 4, 16, 48]        [1, 4, 24, 48]        
[1, 6, 48]            [1, 6, 12, 48]        [1, 6, 12, 24, 48]    [1, 6, 24, 48]        
[1, 8, 48]            [1, 8, 16, 48]        [1, 8, 24, 48]        [1, 12, 48]           
[1, 12, 24, 48]       [1, 16, 48]           [1, 24, 48]           [1, 48]           

48 sequences using second definition:
[2, 24]           [2, 2, 12]        [2, 2, 2, 6]      [2, 2, 2, 2, 3]   
[2, 2, 2, 3, 2]   [2, 2, 3, 4]      [2, 2, 3, 2, 2]   [2, 2, 4, 3]      
[2, 2, 6, 2]      [2, 3, 8]         [2, 3, 2, 4]      [2, 3, 2, 2, 2]   
[2, 3, 4, 2]      [2, 4, 6]         [2, 4, 2, 3]      [2, 4, 3, 2]      
[2, 6, 4]         [2, 6, 2, 2]      [2, 8, 3]         [2, 12, 2]        
[3, 16]           [3, 2, 8]         [3, 2, 2, 4]      [3, 2, 2, 2, 2]   
[3, 2, 4, 2]      [3, 4, 4]         [3, 4, 2, 2]      [3, 8, 2]         
[4, 12]           [4, 2, 6]         [4, 2, 2, 3]      [4, 2, 3, 2]      
[4, 3, 4]         [4, 3, 2, 2]      [4, 4, 3]         [4, 6, 2]         
[6, 8]            [6, 2, 4]         [6, 2, 2, 2]      [6, 4, 2]         
[8, 6]            [8, 2, 3]         [8, 3, 2]         [12, 4]           
[12, 2, 2]        [16, 3]           [24, 2]           [48]              

OEIS A163272:
[0, 1, 48, 1280, 2496, 28672, 29808, 454656, 2342912]