Factor-perfect numbers
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].
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.
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
where a is a list of the factors of n, including n, but excluding 1.
- 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 OEIS A074206 [On the maximal order of numbers in the “factorisatio numerorum” problem] [On Some Asymptotic Formulas in The Theory of The "Factorisatio Numerorum"] Wikipedia: Enumerative Combinatorics
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,
Phix
You can run this online here. Note that my routine names of f(), m(), and g() match my detailed understanding of what I'm doing here perfectly.
with javascript_semantics function f(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 f(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 = f(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)) function g(integer n) sequence f = factors(n) integer l = length(f) sequence d = repeat(0,l) for i=l to 1 by -1 do integer k = f[i] d[i] = 1 integer j = i+1 for x=2*k to n by k do while j<=l and f[j]<x do j+=1 end while if j>l then exit end if if f[j]=x then d[i] += d[j] end if end for end for return 1+sum(d) end function atom t = time(), t1 = t+1 integer n = 4 sequence res = {"0","1"} while length(res)<7 do if g(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)})
- 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 7: 0 1 48 1280 2496 28672 29808 (1.2s)
It takes about 6s under p2js, and 4½ minutes to get the eighth, and 2 hours, 23 minutes and 6s to get the ninth (2342912) on desktop/Phix, compared to 13s for 7 and 39:18 for 8 in Julia (9th not tried), and since python took 2:20 for 7 I didn't even bother trying 8.
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,
Wren
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]