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].
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. 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)
C++
#include <algorithm>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <map>
#include <set>
#include <string>
#include <vector>
std::map<uint32_t, uint32_t> cache{ };
std::string vector_to_string(const std::vector<uint32_t>& vec) {
std::string result = "[";
for ( uint32_t i = 0; i < vec.size() - 1; ++i ) {
result += std::to_string(vec[i]) + ", ";
}
result += std::to_string(vec.back()) + "]";
return result;
}
std::vector<uint32_t> divisors(const uint32_t& number) {
std::set<uint32_t> divisors{ };
uint32_t divisor = 1;
while ( divisor * divisor <= number ) {
if ( number % divisor == 0 ) {
divisors.insert(divisor);
divisors.insert(number / divisor);
}
divisor += 1;
}
std::vector<uint32_t> result(divisors.begin(), divisors.end());
return result;
}
std::vector<std::vector<uint32_t>> more_multiples(const std::vector<uint32_t>& to_vec,
const std::vector<uint32_t>& from_vec) {
std::vector<std::vector<uint32_t>> result{ };
for ( const uint32_t& from : from_vec ) {
if ( from > to_vec.back() && from % to_vec.back() == 0 ) {
std::vector<uint32_t> to_vec_copy = to_vec;
to_vec_copy.emplace_back(from);
result.emplace_back(to_vec_copy);
}
}
std::vector<std::vector<uint32_t>> result_copy = result;
for ( const std::vector<uint32_t>& vec : result_copy ) {
for ( const std::vector<uint32_t>& more : more_multiples(vec, from_vec) ) {
result.emplace_back(more);
}
}
return result;
}
uint32_t erdös_factor_count(const uint32_t& number) {
if ( ! cache.contains(number) ) {
uint32_t factorCount = 0;
const std::vector<uint32_t> divs = divisors(number);
for ( uint32_t i = 1; i < divs.size() - 1; ++i ) {
factorCount += erdös_factor_count(number / divs[i]);
}
factorCount += 1;
cache[number] = factorCount;
}
return cache[number];
}
int main() {
const uint32_t test = 48;
const std::vector<uint32_t> singleton(1, 1);
std::vector<std::vector<uint32_t>> multiples = more_multiples(singleton, divisors(test));
std::set<std::vector<uint32_t>> result_one{ };
for ( std::vector<uint32_t> vec : multiples ) {
if ( vec.back() != test ) {
vec.emplace_back(test);
}
result_one.insert(vec);
}
std::cout << result_one.size() << " sequences using the first definition:" << std::endl;
uint32_t count = 0;
for ( const std::vector<uint32_t>& vec : result_one ) {
const std::string vector_string = vector_to_string(vec);
std::cout << vector_string << ( count++ % 4 == 3 ? "\n" : std::string(23 - vector_string.size(), ' ') );
}
std::cout << std::endl;
std::cout << result_one.size() << " sequences using the second definition:" << std::endl;
std::set<std::vector<uint32_t>> result_two{ };
for ( std::vector<uint32_t> vec : result_one ) {
for ( uint32_t i = 1; i < vec.size(); ++i ) {
vec[i - 1] = vec[i] / vec[i - 1];
}
vec.pop_back();
result_two.insert(vec);
}
count = 0;
for ( const std::vector<uint32_t>& vec : result_two ) {
const std::string vector_string = vector_to_string(vec);
std::cout << vector_string << ( count++ % 4 == 3 ? "\n" : std::string(23 - vector_string.size(), ' ') );
}
std::cout << std::endl;
std::cout << "OEIS A163272: 0 1 ";
for ( uint32_t n = 2; n < 2'400'000; ++n ) {
if ( erdös_factor_count(n) == n ) {
std::cout << n << " ";
}
}
std::cout << std::endl;
}
- Output:
48 sequences using the 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 the 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
FreeBASIC
Type Sequence
As Long value(0 To 100)
As Integer length
End Type
Function properDivisors(n As Long) As Sequence
Dim As Sequence result
result.length = 0
For i As Long = 1 To n\2
If (n Mod i) = 0 Then
result.value(result.length) = i
result.length += 1
End If
Next
Return result
End Function
Function erdosFactorCount(n As Long) As Long
Static As Long cache(0 To 100000)
If cache(n) > 0 Then Return cache(n)
Dim As Long sum = 1
Dim As Sequence divs = properDivisors(n)
For i As Integer = 1 To divs.length - 1
sum += erdosFactorCount(n \ divs.value(i))
Next
cache(n) = sum
Return sum
End Function
Sub appendSequence(sequences() As Sequence, Byref count As Integer, seq As Sequence)
sequences(count) = seq
count += 1
End Sub
Function moreMultiples(bbase As Sequence, divs As Sequence) As Sequence
Dim As Integer i, j
Dim As Sequence result
result.length = 0
For i = 0 To divs.length - 1
If divs.value(i) > bbase.value(bbase.length - 1) Then
If (divs.value(i) Mod bbase.value(bbase.length - 1)) = 0 Then
result.length = bbase.length + 1
For j = 0 To bbase.length - 1
result.value(j) = bbase.value(j)
Next
result.value(bbase.length) = divs.value(i)
End If
End If
Next
Return result
End Function
Function getRatios(seq As Sequence) As Sequence
Dim result As Sequence
result.length = seq.length - 1
For i As Integer = 1 To seq.length - 1
result.value(i-1) = seq.value(i) \ seq.value(i-1)
Next
Return result
End Function
' Main program
Dim As Integer i, j, k
Dim As Integer seqCnt = 0
Dim As Sequence sequences(0 To 100)
' Initialize first sequence
sequences(0).value(0) = 1
sequences(0).length = 1
seqCnt = 1
' Generate sequences
Dim As Sequence divs = properDivisors(48)
Dim As Integer currentIdx = 0
While currentIdx < seqCnt
For i = 0 To divs.length - 1
Dim testSeq As Sequence = sequences(currentIdx)
If divs.value(i) > testSeq.value(testSeq.length - 1) Then
If (divs.value(i) Mod testSeq.value(testSeq.length - 1)) = 0 Then
Dim newSeq As Sequence
newSeq.length = testSeq.length + 1
For j = 0 To testSeq.length - 1
newSeq.value(j) = testSeq.value(j)
Next
newSeq.value(testSeq.length) = divs.value(i)
appendSequence(sequences(), seqCnt, newSeq)
End If
End If
Next
currentIdx += 1
Wend
' Add 48 to all sequences
For i = 0 To seqCnt - 1
sequences(i).value(sequences(i).length) = 48
sequences(i).length += 1
Next
Print seqCnt; " sequences using first definition:"
For i = 0 To seqCnt - 1 Step 3
For j = 0 To 2
If i + j < seqCnt Then
For k = 0 To sequences(i+j).length - 1
Print sequences(i+j).value(k);
If k < sequences(i+j).length - 1 Then Print " ";
Next
Print Space(20 - (sequences(i+j).length * 3));
End If
Next
Print
Next
Print
Print seqCnt; " sequences using second definition:"
For i = 0 To seqCnt - 1 Step 3
For j = 0 To 2
If i + j < seqCnt Then
Dim ratios As Sequence = getRatios(sequences(i+j))
For k = 0 To ratios.length - 1
Print ratios.value(k);
If k < ratios.length - 1 Then Print " ";
Next
Print Space(20 - (ratios.length * 3));
End If
Next
Print
Next
Print !"\nOEIS A163272: [0, 1";
Dim As Integer cnt = 0
Dim As Long n, v
For n = 1 To 10000
v = 4 * n
If erdosFactorCount(v) = v Then
Print ", "; v;
cnt += 1
If cnt = 5 Then Exit For
End If
Next
Print "]"
Sleep
- 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 3 6 48 1 3 12 48 1 3 24 48 1 4 8 48 1 4 12 48 1 4 16 48 1 4 24 48 1 6 12 48 1 6 24 48 1 8 16 48 1 8 24 48 1 12 24 48 1 2 4 8 48 1 2 4 12 48 1 2 4 16 48 1 2 4 24 48 1 2 6 12 48 1 2 6 24 48 1 2 8 16 48 1 2 8 24 48 1 2 12 24 48 1 3 6 12 48 1 3 6 24 48 1 3 12 24 48 1 4 8 16 48 1 4 8 24 48 1 4 12 24 48 1 6 12 24 48 1 2 4 8 16 48 1 2 4 8 24 48 1 2 4 12 24 48 1 2 6 12 24 48 1 3 6 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 3 2 8 3 4 4 3 8 2 4 2 6 4 3 4 4 4 3 4 6 2 6 2 4 6 4 2 8 2 3 8 3 2 12 2 2 2 2 2 6 2 2 3 4 2 2 4 3 2 2 6 2 2 3 2 4 2 3 4 2 2 4 2 3 2 4 3 2 2 6 2 2 3 2 2 4 3 2 4 2 3 4 2 2 4 2 2 3 4 2 3 2 4 3 2 2 6 2 2 2 2 2 2 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 2 2 OEIS A163272: [0, 1, 48, 1280, 2496, 28672, 29808]
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
Java
import java.util.ArrayList;
import java.util.Comparator;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Set;
import java.util.TreeSet;
public final class FactorPerfectNumbers {
public static void main(String[] args) {
final int test = 48;
List<Integer> singleton = new ArrayList<Integer>();
singleton.addLast(1);
List<List<Integer>> multiples = moreMultiples(singleton, divisors(test));
Set<List<Integer>> result = new TreeSet<List<Integer>>(customComparator);
for ( List<Integer> list : multiples ) {
if ( list.getLast() != test ) {
list.addLast(test);
}
result.add(list);
}
System.out.println(result.size() + " sequences using the first definition:");
int count = 0;
for ( List<Integer> list : result ) {
System.out.print(String.format("%-23s%s", list, ( count++ % 4 == 3 ? "\n" : " " )));
}
System.out.println();
System.out.println(result.size() + " sequences using the second definition:");
for ( List<Integer> list : result ) {
for ( int i = 1; i < list.size(); i++ ) {
list.set(i - 1, list.get(i) / list.get(i - 1));
}
list.removeLast();
}
count = 0;
for ( List<Integer> list : result ) {
System.out.print(String.format("%-23s%s", list, ( count++ % 4 == 3 ? "\n" : " " )));
}
System.out.println();
System.out.print("OEIS A163272: 0 1 ");
for ( int n = 2; n < 2_400_000; n++ ) {
if ( erdösFactorCount(n) == n ) {
System.out.print(n + " ");
}
}
System.out.println();
}
private static int erdösFactorCount(int number) {
if ( ! cache.containsKey(number) ) {
int factorCount = 0;
List<Integer> divisors = divisors(number);
for ( int i = 1; i < divisors.size() - 1; i++ ) {
factorCount += erdösFactorCount(number / divisors.get(i));
}
factorCount += 1;
cache.put(number, factorCount);
}
return cache.get(number);
}
private static List<List<Integer>> moreMultiples(List<Integer> toList, List<Integer> fromList) {
List<List<Integer>> result = new ArrayList<List<Integer>>();
for ( int from : fromList ) {
if ( from > toList.getLast() && from % toList.getLast() == 0 ) {
List<Integer> toListCopy = new ArrayList<Integer>(toList);
toListCopy.addLast(from);
result.add(toListCopy);
}
}
List<List<Integer>> resultCopy = new ArrayList<List<Integer>>(result);
for ( List<Integer> list : resultCopy ) {
for ( List<Integer> more : moreMultiples(list, fromList) ) {
result.add(more);
}
}
return result;
}
private static List<Integer> divisors(int number) {
Set<Integer> divisors = new TreeSet<Integer>();
int divisor = 1;
while ( divisor * divisor <= number ) {
if ( number % divisor == 0 ) {
divisors.add(divisor);
divisors.add(number / divisor);
}
divisor += 1;
}
return new ArrayList<Integer>(divisors);
}
private static Comparator<List<Integer>> customComparator = (one, two) -> {
for ( int i = 0; i < Math.min(one.size(), two.size()); i++ ) {
final int comparison = Integer.compare(one.get(i), two.get(i));
if ( comparison != 0 ) {
return comparison;
}
}
return Integer.compare(one.size(), two.size());
};
private static Map<Integer, Integer> cache = new HashMap<Integer, Integer>();
}
- Output:
48 sequences using the 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 the 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
jq
Adapted from Wren
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
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
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; $n += 4 }
say "OEIS A163272: @fpns";
- Output:
48 sequences using first definition: 1 48 1 2 48 1 24 48 1 3 48 1 16 48 1 4 48 1 12 48 1 6 48 1 8 48 1 2 24 48 1 2 16 48 1 2 4 48 1 2 12 48 1 2 6 48 1 2 8 48 1 2 4 24 48 1 2 4 16 48 1 2 4 12 48 1 2 4 8 48 1 2 4 12 24 48 1 2 4 8 24 48 1 2 4 8 16 48 1 2 12 24 48 1 2 6 24 48 1 2 6 12 48 1 2 6 12 24 48 1 2 8 24 48 1 2 8 16 48 1 3 24 48 1 3 12 48 1 3 6 48 1 3 12 24 48 1 3 6 24 48 1 3 6 12 48 1 3 6 12 24 48 1 4 24 48 1 4 16 48 1 4 12 48 1 4 8 48 1 4 12 24 48 1 4 8 24 48 1 4 8 16 48 1 12 24 48 1 6 24 48 1 6 12 48 1 6 12 24 48 1 8 24 48 1 8 16 48 48 sequences using second definition: 48 2 24 24 2 3 16 16 3 4 12 12 4 6 8 8 6 2 12 2 2 8 3 2 2 12 2 6 4 2 3 8 2 4 6 2 2 6 2 2 2 4 3 2 2 3 4 2 2 2 6 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 2 6 2 2 2 3 4 2 2 3 2 4 2 3 2 2 2 2 4 3 2 2 4 2 3 3 8 2 3 4 4 3 2 8 3 4 2 2 3 2 4 2 3 2 2 4 3 2 2 2 2 4 6 2 4 4 3 4 3 4 4 2 6 4 3 2 2 4 2 3 2 4 2 2 3 12 2 2 6 4 2 6 2 4 6 2 2 2 8 3 2 8 2 3 OEIS A163272: 0 1 48 1280 2496 28672 29808
Phix
You can run this online here (expect a blank screen for ~30s).
-- -- demo/rosetta/factor-perfect_numbers.exw -- with javascript_semantics function get_factor_set(integer x) if x=1 then return {1} end if sequence res = {} for k=1 to x-1 do if remainder(x,k)=0 then for y in get_factor_set(k) do res = append(res,y&x) end for end if end for res = sort(res) return res end function function m(sequence s, integer f) sequence res = {} for x in s do x = deep_copy(x) if x[$]!=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
# 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
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
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]