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# Ruth-Aaron numbers

Ruth-Aaron numbers
You are encouraged to solve this task according to the task description, using any language you may know.

A Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime divisors of each integer are equal. So called because 714 is Babe Ruth's lifetime home run record; Hank Aaron's 715th home run broke this record and 714 and 715 have the same prime divisor sum.

A Ruth–Aaron triple consists of three consecutive integers with the same properties.

There is a second variant of Ruth–Aaron numbers, one which uses prime factors rather than prime divisors. The difference; divisors are unique, factors may be repeated. The 714, 715 pair appears in both, so the name still fits.

It is common to refer to each Ruth–Aaron group by the first number in it.

• Find and show, here on this page, the first 30 Ruth-Aaron numbers (factors).
• Find and show, here on this page, the first 30 Ruth-Aaron numbers (divisors).

Stretch
• Find and show the first Ruth-Aaron triple (factors).
• Find and show the first Ruth-Aaron triple (divisors).

## ALGOL 68

Uses sieves for the prime factor sums and prime divisor sums, assumes that the first Ruth-Aaron triples are under 99 000 000.
This uses a large amount of memory - too much for Algol 68G under Windows (and possibly under Linux).
With max number set to 1 000 000, Algol 68G can find the first triple using factors in a few seconds (the loop to find the first divisors triple must be commented out or removed) - Real time: 0.941 s on TIO.RUN for the cutdown version.

BEGIN # find Ruth-Aaron pairs - pairs of consecutive integers where the sum #
# of the prime factors or divisors are equal #
INT max number = 99 000 000; # max number we will consider #
# construct a sieve of primes up to max number #
[ 1 : max number ]BOOL prime;
prime[ 1 ] := FALSE;
prime[ 2 ] := TRUE;
FOR i FROM 3 BY 2 TO UPB prime DO prime[ i ] := TRUE OD;
FOR i FROM 4 BY 2 TO UPB prime DO prime[ i ] := FALSE OD;
FOR i FROM 3 BY 2 TO ENTIER sqrt( UPB prime ) DO
IF prime[ i ] THEN
FOR s FROM i * i BY i + i TO UPB prime DO prime[ s ] := FALSE OD
FI
OD;
# construct the sums of prime divisors up to max number #
[ 1 : max number ]INT ps; FOR n TO max number DO ps[ n ] := 0 OD;
FOR n TO max number DO
IF prime[ n ] THEN
FOR j FROM n BY n TO max number DO ps[ j ] PLUSAB n OD
FI
OD;
INT max count = 30;
# first max count Ruth-Aaron (divisors) numbers #
[ 1 : max count ]INT dra;
INT count := 0;
INT prev sum := 0;
FOR n FROM 2 WHILE count < max count DO
INT this sum = ps[ n ];
IF prev sum = this sum THEN
# found another Ruth-Aaron number #
count PLUSAB 1;
IF count <= max count THEN dra[ count ] := n - 1 FI
FI;
prev sum := this sum
OD;
# first triple #
INT dra3 := 0;
INT pprev sum := ps[ 1 ];
prev sum := ps[ 2 ];
FOR n FROM 3 WHILE dra3 = 0 DO
INT this sum = ps[ n ];
IF prev sum = this sum THEN
IF pprev sum = this sum THEN
# found a Ruth-Aaron triple #
dra3 := n - 2
FI
FI;
pprev sum := prev sum;
prev sum := this sum
OD;
# replace ps with the prime factor count #
INT root max number = ENTIER sqrt( max number );
FOR n FROM 2 TO root max number DO
IF prime[ n ] THEN
INT p := n * n;
WHILE p < root max number DO
FOR j FROM p BY p TO max number DO ps[ j ] PLUSAB n OD;
p TIMESAB n
OD
FI
OD;
# first max count Ruth-Aaron (factors) numbers #
[ 1 : max count ]INT fra;
prev sum := ps[ 1 ];
count := 0;
FOR n FROM 2 WHILE count < 30 DO
INT this sum = ps[ n ];
IF prev sum = this sum THEN
# found another Ruth-Aaron number #
count PLUSAB 1;
fra[ count ] := n - 1
FI;
prev sum := this sum
OD;
# first triple #
prev sum := 0;
count := 0;
INT fra3 := 0;
FOR n FROM 2 WHILE fra3 = 0 DO
INT this sum = ps[ n ];
IF prev sum = this sum AND pprev sum = this sum THEN
# found a Ruth-Aaron triple #
fra3 := n - 2
FI;
pprev sum := prev sum;
prev sum := this sum
OD;
# show the numbers #
print( ( "The first ", whole( max count, 0 ), " Ruth-Aaron numbers (factors):", newline ) );
FOR n TO max count DO
print( ( whole( fra[ n ], - 6 ) ) );
IF n MOD 10 = 0 THEN print( ( newline ) ) FI
OD;
# divisors #
print( ( "The first ", whole( max count, 0 ), " Ruth-Aaron numbers (divisors):", newline ) );
FOR n TO max count DO
print( ( whole( dra[ n ], - 6 ) ) );
IF n MOD 10 = 0 THEN print( ( newline ) ) FI
OD;
# triples #
print( ( newline, "First Ruth-Aaron triple (factors): ", whole( fra3, 0 ) ) );
print( ( newline, "First Ruth-Aaron triple (divisors): ", whole( dra3, 0 ) ) )
END
Output:
The first 30 Ruth-Aaron numbers (factors):
5     8    15    77   125   714   948  1330  1520  1862
2491  3248  4185  4191  5405  5560  5959  6867  8280  8463
10647 12351 14587 16932 17080 18490 20450 24895 26642 26649
The first 30 Ruth-Aaron numbers (divisors):
5    24    49    77   104   153   369   492   714  1682
2107  2299  2600  2783  5405  6556  6811  8855  9800 12726
13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

First Ruth-Aaron triple (factors):  417162
First Ruth-Aaron triple (divisors): 89460294

## C++

This takes about 2 minutes 24 seconds (3.2GHz Quad-Core Intel Core i5).

#include <iomanip>
#include <iostream>

int prime_factor_sum(int n) {
int sum = 0;
for (; (n & 1) == 0; n >>= 1)
sum += 2;
for (int p = 3, sq = 9; sq <= n; p += 2) {
for (; n % p == 0; n /= p)
sum += p;
sq += (p + 1) << 2;
}
if (n > 1)
sum += n;
return sum;
}

int prime_divisor_sum(int n) {
int sum = 0;
if ((n & 1) == 0) {
sum += 2;
n >>= 1;
while ((n & 1) == 0)
n >>= 1;
}
for (int p = 3, sq = 9; sq <= n; p += 2) {
if (n % p == 0) {
sum += p;
n /= p;
while (n % p == 0)
n /= p;
}
sq += (p + 1) << 2;
}
if (n > 1)
sum += n;
return sum;
}

int main() {
const int limit = 30;
int dsum1 = 0, fsum1 = 0, dsum2 = 0, fsum2 = 0;

std::cout << "First " << limit << " Ruth-Aaron numbers (factors):\n";
for (int n = 2, count = 0; count < limit; ++n) {
fsum2 = prime_factor_sum(n);
if (fsum1 == fsum2) {
++count;
std::cout << std::setw(5) << n - 1
<< (count % 10 == 0 ? '\n' : ' ');
}
fsum1 = fsum2;
}

std::cout << "\nFirst " << limit << " Ruth-Aaron numbers (divisors):\n";
for (int n = 2, count = 0; count < limit; ++n) {
dsum2 = prime_divisor_sum(n);
if (dsum1 == dsum2) {
++count;
std::cout << std::setw(5) << n - 1
<< (count % 10 == 0 ? '\n' : ' ');
}
dsum1 = dsum2;
}

dsum1 = 0, fsum1 = 0, dsum2 = 0, fsum2 = 0;
for (int n = 2;; ++n) {
int fsum3 = prime_factor_sum(n);
if (fsum1 == fsum2 && fsum2 == fsum3) {
std::cout << "\nFirst Ruth-Aaron triple (factors): " << n - 2
<< '\n';
break;
}
fsum1 = fsum2;
fsum2 = fsum3;
}
for (int n = 2;; ++n) {
int dsum3 = prime_divisor_sum(n);
if (dsum1 == dsum2 && dsum2 == dsum3) {
std::cout << "\nFirst Ruth-Aaron triple (divisors): " << n - 2
<< '\n';
break;
}
dsum1 = dsum2;
dsum2 = dsum3;
}
}
Output:
First 30 Ruth-Aaron numbers (factors):
5     8    15    77   125   714   948  1330  1520  1862
2491  3248  4185  4191  5405  5560  5959  6867  8280  8463
10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

First 30 Ruth-Aaron numbers (divisors):
5    24    49    77   104   153   369   492   714  1682
2107  2299  2600  2783  5405  6556  6811  8855  9800 12726
13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

First Ruth-Aaron triple (factors): 417162

First Ruth-Aaron triple (divisors): 89460294

## Factor

Works with: Factor version 0.99 2022-04-03
USING: assocs.extras grouping io kernel lists lists.lazy math
math.primes.factors prettyprint ranges sequences ;

: pair-same? ( ... n quot: ( ... m -- ... n ) -- ... ? )
[ dup 1 + ] dip same? ; inline

: RA-f? ( n -- ? ) [ factors sum ] pair-same? ;
: RA-d? ( n -- ? ) [ group-factors sum-keys ] pair-same? ;
: filter-naturals ( quot -- list ) 1 lfrom swap lfilter ; inline
: RA-f ( -- list ) [ RA-f? ] filter-naturals ;
: RA-d ( -- list ) [ RA-d? ] filter-naturals ;

: list. ( list -- )
30 swap ltake list>array 10 group simple-table. ;

"First 30 Ruth-Aaron numbers (factors):" print
RA-f list. nl

"First 30 Ruth-Aaron numbers (divisors):" print
RA-d list.
Output:
First 30 Ruth-Aaron numbers (factors):
5     8     15    77    125   714   948   1330  1520  1862
2491  3248  4185  4191  5405  5560  5959  6867  8280  8463
10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

First 30 Ruth-Aaron numbers (divisors):
5     24    49    77    104   153   369   492   714   1682
2107  2299  2600  2783  5405  6556  6811  8855  9800  12726
13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

## Go

Translation of: Wren
Library: Go-rcu

package main

import (
"fmt"
"rcu"
)

func prune(a []int) []int {
prev := a[0]
b := []int{prev}
for i := 1; i < len(a); i++ {
if a[i] != prev {
b = append(b, a[i])
prev = a[i]
}
}
return b
}

func main() {
var resF, resD, resT, factors1 []int
factors2 := []int{2}
factors3 := []int{3}
var sum1, sum2, sum3 int = 0, 2, 3
var countF, countD, countT int
for n := 2; countT < 1 || countD < 30 || countF < 30; n++ {
factors1 = factors2
factors2 = factors3
factors3 = rcu.PrimeFactors(n + 2)
sum1 = sum2
sum2 = sum3
sum3 = rcu.SumInts(factors3)
if countF < 30 && sum1 == sum2 {
resF = append(resF, n)
countF++
}
if sum1 == sum2 && sum2 == sum3 {
resT = append(resT, n)
countT++
}
if countD < 30 {
factors4 := make([]int, len(factors1))
copy(factors4, factors1)
factors5 := make([]int, len(factors2))
copy(factors5, factors2)
factors4 = prune(factors4)
factors5 = prune(factors5)
if rcu.SumInts(factors4) == rcu.SumInts(factors5) {
resD = append(resD, n)
countD++
}
}
}
fmt.Println("First 30 Ruth-Aaron numbers (factors):")
fmt.Println(resF)
fmt.Println("\nFirst 30 Ruth-Aaron numbers (divisors):")
fmt.Println(resD)
fmt.Println("\nFirst Ruth-Aaron triple (factors):")
fmt.Println(resT[0])

resT = resT[:0]
factors1 = factors1[:0]
factors2 = factors2[:1]
factors2[0] = 2
factors3 = factors3[:1]
factors3[0] = 3
countT = 0
for n := 2; countT < 1; n++ {
factors1 = factors2
factors2 = factors3
factors3 = prune(rcu.PrimeFactors(n + 2))
sum1 = sum2
sum2 = sum3
sum3 = rcu.SumInts(factors3)
if sum1 == sum2 && sum2 == sum3 {
resT = append(resT, n)
countT++
}
}
fmt.Println("\nFirst Ruth-Aaron triple (divisors):")
fmt.Println(resT[0])
}
Output:
First 30 Ruth-Aaron numbers (factors):
[5 8 15 77 125 714 948 1330 1520 1862 2491 3248 4185 4191 5405 5560 5959 6867 8280 8463 10647 12351 14587 16932 17080 18490 20450 24895 26642 26649]

First 30 Ruth-Aaron numbers (divisors):
[5 24 49 77 104 153 369 492 714 1682 2107 2299 2600 2783 5405 6556 6811 8855 9800 12726 13775 18655 21183 24024 24432 24880 25839 26642 35456 40081]

First Ruth-Aaron triple (factors):
417162

First Ruth-Aaron triple (divisors):
89460294

import qualified Data.Set as S
import Data.List.Split ( chunksOf )

divisors :: Int -> [Int]
divisors n = [d | d <- [2 .. n] , mod n d == 0]

--for obvious theoretical reasons the smallest divisor of a number bare 1
--must be prime
primeFactors :: Int -> [Int]
primeFactors n = snd \$ until ( (== 1) . fst ) step (n , [] )
where
step :: (Int , [Int] ) -> (Int , [Int] )
step (n , li) = ( div n h , li ++ [h] )
where
h :: Int
h = head \$ divisors n

primeDivisors :: Int -> [Int]
primeDivisors n = S.toList \$ S.fromList \$ primeFactors n

solution :: (Int -> [Int] ) -> [Int]
solution f = snd \$ until ( (== 30 ) . length . snd ) step ([2 , 3] , [] )
where
step :: ([Int] , [Int] ) -> ([Int] , [Int])
step ( neighbours , ranums ) = ( map ( + 1 ) neighbours , if (sum \$ f
\$ head neighbours ) == (sum \$ f \$ last neighbours) then
ranums ++ [ head neighbours ] else ranums )

formatNumber :: Int -> String -> String
formatNumber width num
|width > l = replicate ( width -l ) ' ' ++ num
|width == l = num
|width < l = num
where
l = length num

main :: IO ( )
main = do
let ruth_aaron_pairs = solution primeFactors
maxlen = length \$ show \$ last ruth_aaron_pairs
numberlines = chunksOf 8 \$ map show ruth_aaron_pairs
ruth_aaron_divisors = solution primeDivisors
maxlen2 = length \$ show \$ last ruth_aaron_divisors
numberlines2 = chunksOf 8 \$ map show ruth_aaron_divisors
putStrLn "First 30 Ruth-Aaaron numbers ( factors ) :"
mapM_ (\nlin -> putStrLn \$ foldl1 ( ++ ) \$ map (\st -> formatNumber (maxlen + 2) st )
nlin ) numberlines
putStrLn " "
putStrLn "First 30 Ruth-Aaron numbers( divisors ):"
mapM_ (\nlin -> putStrLn \$ foldl1 ( ++ ) \$ map (\st -> formatNumber (maxlen2 + 2) st )
nlin ) numberlines2
Output:
First 30 Ruth-Aaaron numbers ( factors ) :
5      8     15     77    125    714    948   1330
1520   1862   2491   3248   4185   4191   5405   5560
5959   6867   8280   8463  10647  12351  14587  16932
17080  18490  20450  24895  26642  26649

First 30 Ruth-Aaron numbers( divisors ):
5     24     49     77    104    153    369    492
714   1682   2107   2299   2600   2783   5405   6556
6811   8855   9800  12726  13775  18655  21183  24024
24432  24880  25839  26642  35456  40081

## J

Thus:

NB. using factors
30{.1 2+/~I. 2 =/\ +/@q: 1+i.100000
5 6
8 9
15 16
77 78
125 126
714 715
948 949
1330 1331
1520 1521
1862 1863
2491 2492
3248 3249
4185 4186
4191 4192
5405 5406
5560 5561
5959 5960
6867 6868
8280 8281
8463 8464
10647 10648
12351 12352
14587 14588
16932 16933
17080 17081
18490 18491
20450 20451
24895 24896
26642 26643
26649 26650

NB. using divisors
30{.1 2+/~I. 2 =/\ (+/@{[email protected]:~&__) 1+i.100000
5 6
24 25
49 50
77 78
104 105
153 154
369 370
492 493
714 715
1682 1683
2107 2108
2299 2300
2600 2601
2783 2784
5405 5406
6556 6557
6811 6812
8855 8856
9800 9801
12726 12727
13775 13776
18655 18656
21183 21184
24024 24025
24432 24433
24880 24881
25839 25840
26642 26643
35456 35457
40081 40082

## Julia

using Lazy
using Primes

sumprimedivisors(n) = sum([p[1] for p in factor(n)])
ruthaaron(n) = sumprimedivisors(n) == sumprimedivisors(n + 1)
ruthaarontriple(n) = sumprimedivisors(n) == sumprimedivisors(n + 1) ==
sumprimedivisors(n + 2)

sumprimefactors(n) = sum([p[1] * p[2] for p in factor(n)])
ruthaaronfactors(n) = sumprimefactors(n) == sumprimefactors(n + 1)
ruthaaronfactorstriple(n) = sumprimefactors(n) == sumprimefactors(n + 1) ==
sumprimefactors(n + 2)

raseq = @>> Lazy.range() filter(ruthaaron)
rafseq = @>> Lazy.range() filter(ruthaaronfactors)

println("30 Ruth Aaron numbers:")
foreach(p -> print(lpad(p[2], 6), p[1] % 10 == 0 ? "\n" : ""),
enumerate(collect(take(30, raseq))))

println("\n30 Ruth Aaron factor numbers:")
foreach(p -> print(lpad(p[2], 6), p[1] % 10 == 0 ? "\n" : ""),
enumerate(collect(take(30, rafseq))))

println("\nRuth Aaron triple starts at: ", findfirst(ruthaarontriple, 1:100000000))
println("\nRuth Aaron factor triple starts at: ", findfirst(ruthaaronfactorstriple, 1:10000000))

Output:
30 Ruth Aaron numbers:
5    24    49    77   104   153   369   492   714  1682
2107  2299  2600  2783  5405  6556  6811  8855  9800 12726
13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

30 Ruth Aaron factor numbers:
5     8    15    77   125   714   948  1330  1520  1862
2491  3248  4185  4191  5405  5560  5959  6867  8280  8463
10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

Ruth Aaron triple starts at: 89460294

Ruth Aaron factor triple starts at: 417162

## Pascal

### Free Pascal

all depends on fast prime decomposition.

program RuthAaronNumb;
// gets factors of consecutive integers fast
// limited to 1.2e11
{\$IFDEF FPC}
{\$MODE DELPHI} {\$OPTIMIZATION ON,ALL} {\$COPERATORS ON}
{\$ELSE}
{\$APPTYPE CONSOLE}
{\$ENDIF}
uses
sysutils,
strutils //Numb2USA
{\$IFDEF WINDOWS},Windows{\$ENDIF}
;
//######################################################################
//prime decomposition
const
//HCN(86) > 1.2E11 = 128,501,493,120 count of divs = 4096 7 3 1 1 1 1 1 1 1
HCN_DivCnt = 4096;
//used odd size for test only
SizePrDeFe = 32768;//*72 <= 64kb level I or 2 Mb ~ level 2 cache
type
tItem = Uint64;
tDivisors = array [0..HCN_DivCnt] of tItem;
tpDivisor = pUint64;

tdigits = array [0..31] of Uint32;
//the first number with 11 different prime factors =
//2*3*5*7*11*13*17*19*23*29*31 = 2E11
//56 byte
tprimeFac = packed record
pfSumOfDivs,
pfRemain : Uint64;
pfDivCnt : Uint32;
pfMaxIdx : Uint32;
pfpotPrimIdx : array[0..9] of word;
pfpotMax : array[0..11] of byte;
end;
tpPrimeFac = ^tprimeFac;

tPrimeDecompField = array[0..SizePrDeFe-1] of tprimeFac;
tPrimes = array[0..65535] of Uint32;

var
{\$ALIGN 8}
SmallPrimes: tPrimes;
{\$ALIGN 32}
PrimeDecompField :tPrimeDecompField;
pdfIDX,pdfOfs: NativeInt;

procedure InitSmallPrimes;
//get primes. #0..65535.Sieving only odd numbers
const
MAXLIMIT = (821641-1) shr 1;
var
pr : array[0..MAXLIMIT] of byte;
p,j,d,flipflop :NativeUInt;
Begin
SmallPrimes[0] := 2;
fillchar(pr[0],SizeOf(pr),#0);
p := 0;
repeat
repeat
p +=1
until pr[p]= 0;
j := (p+1)*p*2;
if j>MAXLIMIT then
BREAK;
d := 2*p+1;
repeat
pr[j] := 1;
j += d;
until j>MAXLIMIT;
until false;

SmallPrimes[1] := 3;
SmallPrimes[2] := 5;
j := 3;
d := 7;
flipflop := (2+1)-1;//7+2*2,11+2*1,13,17,19,23
p := 3;
repeat
if pr[p] = 0 then
begin
SmallPrimes[j] := d;
inc(j);
end;
d += 2*flipflop;
p+=flipflop;
flipflop := 3-flipflop;
until (p > MAXLIMIT) OR (j>High(SmallPrimes));
end;

function OutPots(pD:tpPrimeFac;n:NativeInt):Ansistring;
var
s: String[31];
chk,p,i: NativeInt;
Begin
str(n,s);
result := Format('%15s : ',[Numb2USA(s)]);

with pd^ do
begin
chk := 1;
For n := 0 to pfMaxIdx-1 do
Begin
if n>0 then
result += '*';
p := SmallPrimes[pfpotPrimIdx[n]];
chk *= p;
str(p,s);
result += s;
i := pfpotMax[n];
if i >1 then
Begin
str(pfpotMax[n],s);
result += '^'+s;
repeat
chk *= p;
dec(i);
until i <= 1;
end;
end;
p := pfRemain;
If p >1 then
Begin
str(p,s);
chk *= p;
result += '*'+s;
end;
end;
end;

function CnvtoBASE(var dgt:tDigits;n:Uint64;base:NativeUint):NativeInt;
//n must be multiple of base aka n mod base must be 0
var
q,r: Uint64;
i : NativeInt;
Begin
fillchar(dgt,SizeOf(dgt),#0);
i := 0;
n := n div base;
result := 0;
repeat
r := n;
q := n div base;
r -= q*base;
n := q;
dgt[i] := r;
inc(i);
until (q = 0);
//searching lowest pot in base
result := 0;
while (result<i) AND (dgt[result] = 0) do
inc(result);
inc(result);
end;

function IncByBaseInBase(var dgt:tDigits;base:NativeInt):NativeInt;
var
q :NativeInt;
Begin
result := 0;
q := dgt[result]+1;
if q = base then
repeat
dgt[result] := 0;
inc(result);
q := dgt[result]+1;
until q <> base;
dgt[result] := q;
result +=1;
end;

function SieveOneSieve(var pdf:tPrimeDecompField):boolean;
var
dgt:tDigits;
i,j,k,pr,fac,n,MaxP : Uint64;
begin
n := pdfOfs;
if n+SizePrDeFe >= sqr(SmallPrimes[High(SmallPrimes)]) then
EXIT(FALSE);
//init
for i := 0 to SizePrDeFe-1 do
begin
with pdf[i] do
Begin
pfDivCnt := 1;
pfSumOfDivs := 1;
pfRemain := n+i;
pfMaxIdx := 0;
pfpotPrimIdx[0] := 0;
pfpotMax[0] := 0;
end;
end;
//first factor 2. Make n+i even
i := (pdfIdx+n) AND 1;
IF (n = 0) AND (pdfIdx<2) then
i := 2;

repeat
with pdf[i] do
begin
j := BsfQWord(n+i);
pfMaxIdx := 1;
pfpotPrimIdx[0] := 0;
pfpotMax[0] := j;
pfRemain := (n+i) shr j;
pfSumOfDivs := (Uint64(1) shl (j+1))-1;
pfDivCnt := j+1;
end;
i += 2;
until i >=SizePrDeFe;
//i now index in SmallPrimes
i := 0;
maxP := trunc(sqrt(n+SizePrDeFe))+1;
repeat
//search next prime that is in bounds of sieve
if n = 0 then
begin
repeat
inc(i);
pr := SmallPrimes[i];
k := pr-n MOD pr;
if k < SizePrDeFe then
break;
until pr > MaxP;
end
else
begin
repeat
inc(i);
pr := SmallPrimes[i];
k := pr-n MOD pr;
if (k = pr) AND (n>0) then
k:= 0;
if k < SizePrDeFe then
break;
until pr > MaxP;
end;

//no need to use higher primes
if pr*pr > n+SizePrDeFe then
BREAK;

//j is power of prime
j := CnvtoBASE(dgt,n+k,pr);
repeat
with pdf[k] do
Begin
pfpotPrimIdx[pfMaxIdx] := i;
pfpotMax[pfMaxIdx] := j;
pfDivCnt *= j+1;
fac := pr;
repeat
pfRemain := pfRemain DIV pr;
dec(j);
fac *= pr;
until j<= 0;
pfSumOfDivs *= (fac-1)DIV(pr-1);
inc(pfMaxIdx);
k += pr;
j := IncByBaseInBase(dgt,pr);
end;
until k >= SizePrDeFe;
until false;

//correct sum of & count of divisors
for i := 0 to High(pdf) do
Begin
with pdf[i] do
begin
j := pfRemain;
if j <> 1 then
begin
pfSumOFDivs *= (j+1);
pfDivCnt *=2;
end;
end;
end;
result := true;
end;

function NextSieve:boolean;
begin
dec(pdfIDX,SizePrDeFe);
inc(pdfOfs,SizePrDeFe);
result := SieveOneSieve(PrimeDecompField);
end;

function GetNextPrimeDecomp:tpPrimeFac;
begin
if pdfIDX >= SizePrDeFe then
if Not(NextSieve) then
EXIT(NIL);
result := @PrimeDecompField[pdfIDX];
inc(pdfIDX);
end;

function Init_Sieve(n:NativeUint):boolean;
//Init Sieve pdfIdx,pdfOfs are Global
begin
pdfIdx := n MOD SizePrDeFe;
pdfOfs := n-pdfIdx;
result := SieveOneSieve(PrimeDecompField);
end;
//end prime decomposition
//######################################################################

procedure Get_RA_Prime(cntlimit:NativeUInt;useFactors:Boolean);
var
pPrimeDecomp :tpPrimeFac;
pr,sum0,sum1,n,i,cnt : NativeUInt;
begin
write('First 30 Ruth-Aaron numbers (');
if useFactors then
writeln('factors ):')
else
writeln('divisors ):');

cnt := 0;
sum1:= 0;
n := 2;
Init_Sieve(n);
repeat
pPrimeDecomp:= GetNextPrimeDecomp;
with pPrimeDecomp^ do
begin
sum0:= pfRemain;
//if not(prime)
if (sum0 <> n) then
begin
if sum0 = 1 then
sum0 := 0;
For i := 0 to pfMaxIdx-1 do
begin
pr := smallprimes[pfpotPrimIdx[i]];
if useFactors then
sum0 += pr*pfpotMax[i]
else
sum0 += pr;
end;
if sum1 = sum0 then
begin
write(n-1:10);
inc(cnt);
if cnt mod 8 = 0 then
writeln;
end;
sum1 := sum0;
end
else
sum1:= 0;
end;
inc(n);
until cnt>=cntlimit;
writeln;
end;

function findfirstTripplesFactor(useFactors:boolean):NativeUint;
var
pPrimeDecomp :tpPrimeFac;
pr,sum0,sum1,sum2,i : NativeUInt;
begin
sum1:= 0;
sum2:= 0;
result:= 2;
Init_Sieve(result);
repeat
pPrimeDecomp:= GetNextPrimeDecomp;
with pPrimeDecomp^ do
begin
sum0:= pfRemain;
//if not(prime)
if (sum0 <> result) then
begin
if sum0 = 1 then
sum0 := 0;
For i := 0 to pfMaxIdx-1 do
begin
pr := smallprimes[pfpotPrimIdx[i]];
if useFactors then
pr *= pfpotMax[i];
sum0 += pr
end;
if (sum2 = sum0) AND (sum1=sum0) then
Exit(result-2);
end
else
sum0 := 0;
sum2:= sum1;
sum1 := sum0;
end;
inc(result);
until false
end;

Begin
InitSmallPrimes;
Get_RA_Prime(30,false);
Get_RA_Prime(30,true);
writeln;
writeln('First Ruth-Aaron triple (factors) :');
writeln(findfirstTripplesFactor(true):10);
writeln;
writeln('First Ruth-Aaron triple (divisors):');
writeln(findfirstTripplesFactor(false):10);
end.
@TIO.RUN:
Real time: 6.811 s CPU share: 99.35 %
First 30 Ruth-Aaron numbers (divisors ):
5        24        49        77       104       153       369       492
714      1682      2107      2299      2600      2783      5405      6556
6811      8855      9800     12726     13775     18655     21183     24024
24432     24880     25839     26642     35456     40081
First 30 Ruth-Aaron numbers (factors ):
5         8        15        77       125       714       948      1330
1520      1862      2491      3248      4185      4191      5405      5560
5959      6867      8280      8463     10647     12351     14587     16932
17080     18490     20450     24895     26642     26649

First Ruth-Aaron triple (factors) :
417162

First Ruth-Aaron triple (divisors):
89460294

## Perl

#!/usr/bin/perl

use strict;
use warnings;
use ntheory qw( factor vecsum );
use List::AllUtils qw( uniq );

#use Data::Dump 'dd'; dd factor(6); exit;

my \$n = 1;
{
vecsum(factor(\$n)) == vecsum(factor(\$n+1)) and push @answers, \$n;
\$n++;
}

\$n = 1;
{
vecsum(uniq factor(\$n)) == vecsum(uniq factor(\$n+1)) and push @answers, \$n;
\$n++;
}
Output:
factors:

5 8 15 77 125 714 948 1330 1520 1862 2491 3248 4185 4191 5405
5560 5959 6867 8280 8463 10647 12351 14587 16932 17080 18490
20450 24895 26642 26649

divisors:

5 24 49 77 104 153 369 492 714 1682 2107 2299 2600 2783 5405
6556 6811 8855 9800 12726 13775 18655 21183 24024 24432 24880
25839 26642 35456 40081

## Phix

Library: Phix/online

You can run this online here.

with javascript_semantics
procedure ruth_aaron(bool d, integer n=30, l=2, i=1)
string fd = iff(d?"divisors":"factors"),
ns = iff(n=1?"":sprintf(" %d",n)),
ss = iff(n=1?"":"s"),
nt = iff(l=2?"number":"triple")
printf(1,"First%s Ruth-Aaron %s%s (%s):\n",{ns,nt,ss,fd})
integer prev = -1, k = i, c = 0
while n do
sequence f = prime_factors(k,true,-1)
if d then f = unique(f) end if
integer s = sum(f)
if s and s=prev then
c += 1
if c=l-1 then
printf(1,"%d ",k-c)
n -= 1
end if
else
c = 0
end if
prev = s
k += 1
end while
printf(1,"\n\n")
end procedure
atom t0 = time()
ruth_aaron(false)
ruth_aaron(true)
ruth_aaron(false, 1, 3) -- (2.1s)
-- give this one a little leg-up :-) ...
ruth_aaron(true, 1, 3, 89460000) -- (0.1s)
--ruth_aaron(true, 1, 3) -- (24 minutes 30s)
Output:
First 30 Ruth-Aaron numbers (factors):
5 8 15 77 125 714 948 1330 1520 1862 2491 3248 4185 4191 5405 5560 5959 6867 8280 8463 10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

First 30 Ruth-Aaron numbers (divisors):
5 24 49 77 104 153 369 492 714 1682 2107 2299 2600 2783 5405 6556 6811 8855 9800 12726 13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

First Ruth-Aaron triple (factors):
417162

First Ruth-Aaron triple (divisors):
89460294

## Quackery

primefactors is defined at Prime decomposition#Quackery.

[ behead dup dip nested rot
witheach
[ tuck != if
[ dup dip
[ nested join ] ] ]
drop ] is -duplicates ( [ --> [ )

[ primefactors -duplicates ] is primedivisors ( n --> n )

[ 0 swap witheach + ] is sum ( [ --> n )

[ [] temp put
3 2 primefactors sum
[ over primefactors sum
tuck = if
[ over 1 -
temp take
swap join
temp put ]
dip 1+
temp share size 30 = until ]
2drop
temp take ] is raf ( --> )

[ [] temp put
3 2 primedivisors sum
[ over primedivisors sum
tuck = if
[ over 1 -
temp take
swap join
temp put ]
dip 1+
temp share size 30 = until ]
2drop
temp take ] is rad ( --> )

raf echo
cr cr
Output:
[ 5 8 15 77 125 714 948 1330 1520 1862 2491 3248 4185 4191 5405 5560 5959 6867 8280 8463 10647 12351 14587 16932 17080 18490 20450 24895 26642 26649 ]

[ 5 24 49 77 104 153 369 492 714 1682 2107 2299 2600 2783 5405 6556 6811 8855 9800 12726 13775 18655 21183 24024 24432 24880 25839 26642 35456 40081 ]

## Raku

use Prime::Factor;

my @pf = lazy (^).hyper(:1000batch).map: *.&prime-factors.sum;
my @upf = lazy (^).hyper(:1000batch).map: *.&prime-factors.unique.sum;

put "First 30 Ruth-Aaron numbers (Factors):\n" ~
(1..).grep( { @pf[\$_] == @pf[\$_ + 1] } )[^30];

put "\nFirst 30 Ruth-Aaron numbers (Divisors):\n" ~
(1..).grep( { @upf[\$_] == @upf[\$_ + 1] } )[^30];

# Stretch: ~ 5 seconds
put "\nFirst Ruth-Aaron triple (Factors):\n" ~
(1..).first: { @pf[\$_] == @pf[\$_ + 1] == @pf[\$_ + 2] }

# Really, really, _really_ slow. 186(!) minutes... but with no cheating or "leg up".
put "\nFirst Ruth-Aaron triple (Divisors):\n" ~
(1..).first: { @upf[\$_] == @upf[\$_ + 1] == @upf[\$_ + 2] }
Output:
First 30 Ruth-Aaron numbers (Factors):
5 8 15 77 125 714 948 1330 1520 1862 2491 3248 4185 4191 5405 5560 5959 6867 8280 8463 10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

First 30 Ruth-Aaron numbers (Divisors):
5 24 49 77 104 153 369 492 714 1682 2107 2299 2600 2783 5405 6556 6811 8855 9800 12726 13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

First Ruth-Aaron triple (Factors):
417162

First Ruth-Aaron triple (Divisors):
89460294

## Sidef

say "First 30 Ruth-Aaron numbers (factors):"
say 30.by {|n| (sopfr(n) == sopfr(n+1)) && (n > 0) }.join(' ')

say "\nFirst 30 Ruth-Aaron numbers (divisors):"
say 30.by {|n| ( sopf(n) == sopf(n+1)) && (n > 0) }.join(' ')
Output:
First 30 Ruth-Aaron numbers (factors):
5 8 15 77 125 714 948 1330 1520 1862 2491 3248 4185 4191 5405 5560 5959 6867 8280 8463 10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

First 30 Ruth-Aaron numbers (divisors):
5 24 49 77 104 153 369 492 714 1682 2107 2299 2600 2783 5405 6556 6811 8855 9800 12726 13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

## Wren

Library: Wren-math
Library: Wren-seq
Library: Wren-fmt

To find the first thirty Ruth-Aaron pairs and the first triple based on factors takes around 2.2 seconds.

However, with nearly 90 million trios of numbers to slog through, it takes around 68 minutes to find the first triple based on divisors.

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

var resF = []
var resD = []
var resT = [] // factors only
var n = 2
var factors1 = []
var factors2 = [2]
var factors3 = [3]
var sum1 = 0
var sum2 = 2
var sum3 = 3
var countF = 0
var countD = 0
var countT = 0
while (countT < 1 || countD < 30 || countF < 30) {
factors1 = factors2
factors2 = factors3
factors3 = Int.primeFactors(n+2)
sum1 = sum2
sum2 = sum3
sum3 = Nums.sum(factors3)
if (countF < 30 && sum1 == sum2) {
countF = countF + 1
}
if (sum1 == sum2 && sum2 == sum3) {
countT = countT + 1
}
if (countD < 30) {
var factors4 = factors1.toList
var factors5 = factors2.toList
Lst.prune(factors4)
Lst.prune(factors5)
if (Nums.sum(factors4) == Nums.sum(factors5)) {
countD = countD + 1
}
}
n = n + 1
}

System.print("First 30 Ruth-Aaron numbers (factors):")
System.print(resF.join(" "))
System.print("\nFirst 30 Ruth-Aaron numbers (divisors):")
System.print(resD.join(" "))
System.print("\nFirst Ruth-Aaron triple (factors):")
System.print(resT[0])

resT = [] // divisors only
n = 2
factors1 = []
factors2 = [2]
factors3 = [3]
sum1 = 0
sum2 = 2
sum3 = 3
countT = 0
while (countT < 1) {
factors1 = factors2
factors2 = factors3
factors3 = Int.primeFactors(n+2)
Lst.prune(factors3)
sum1 = sum2
sum2 = sum3
sum3 = Nums.sum(factors3)
if (sum1 == sum2 && sum2 == sum3) {
countT = countT + 1
}
n = n + 1
}

System.print("\nFirst Ruth-Aaron triple (divisors):")
System.print(resT[0])
Output:
First 30 Ruth-Aaron numbers (factors):
5 8 15 77 125 714 948 1330 1520 1862 2491 3248 4185 4191 5405 5560 5959 6867 8280 8463 10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

First 30 Ruth-Aaron numbers (divisors):
5 24 49 77 104 153 369 492 714 1682 2107 2299 2600 2783 5405 6556 6811 8855 9800 12726 13775 18655 21183 24024 24432 24880 25839 26642 35456 40081

First Ruth-Aaron triple (factors):
417162

First Ruth-Aaron triple (divisors):
89460294

## XPL0

func DivSum(N, AllDiv); \Return sum of divisors
int N, AllDiv; \all divisors vs. only prime divisors
int F, F0, S, Q;
[F:= 2; F0:= 0; S:= 0;
repeat Q:= N/F;
if rem(0) = 0 then
[if AllDiv then S:= S+F
else if F # F0 then
[S:= S+F; F0:= F];
N:= Q;
]
else F:= F+1;
until F > N;
return S;
];

proc Ruth(AllDiv); \Show Ruth-Aaron numbers
int AllDiv;
int C, S, S0, N;
[C:= 0; S0:= 0;
N:= 2;
repeat S:= DivSum(N, AllDiv);
if S = S0 then
[IntOut(0, N-1);
C:= C+1;
if rem(C/10) = 0 then CrLf(0) else ChOut(0, ^ );
];
S0:= S;
N:= N+1;
until C >= 30;
];

[Ruth(true);
CrLf(0);
Ruth(false);
]
Output:
5 8 15 77 125 714 948 1330 1520 1862
2491 3248 4185 4191 5405 5560 5959 6867 8280 8463
10647 12351 14587 16932 17080 18490 20450 24895 26642 26649

5 24 49 77 104 153 369 492 714 1682
2107 2299 2600 2783 5405 6556 6811 8855 9800 12726
13775 18655 21183 24024 24432 24880 25839 26642 35456 40081