Pairs with common factors

Pairs with common factors
You are encouraged to solve this task according to the task description, using any language you may know.

Generate the sequence where each term n is the count of the pairs (x,y) with 1 < x < y <= n, that have at least one common prime factor.

For instance, when n = 9, examine the pairs:

```   (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9)
(3,4) (3,5) (3,6) (3,7) (3,8) (3,9)
(4,5) (4,6) (4,7) (4,8) (4,9)
(5,6) (5,7) (5,8) (5,9)
(6,7) (6,8) (6,9)
(7,8) (7,9)
(8,9)
```

Find all of the pairs that have at least one common prime factor:

```   (2,4) (2,6) (2,8) (3,6) (3,9) (4,6) (4,8) (6,8) (6,9)
```

and count them:

```   a(9) = 9
```

Terms may be found directly using the formula:

```   a(n) = n × (n-1) / 2 + 1 - 𝚺{i=1..n} 𝚽(i)
```

where 𝚽() is Phi; the Euler totient function.

For the term a(p), if p is prime, then a(p) is equal to the previous term.

• Find and display the first one hundred terms of the sequence.
• Find and display the one thousandth.

Stretch
• Find and display the ten thousandth, one hundred thousandth, one millionth.

ALGOL 68

Works with: ALGOL 68G version 3 - tested with release 3.0.3.win32

Should work with other implementations of Algol 68 where LONG INT is at least 64 bits. Unfortunately, Algol 68G version 2 runs out of memory sometime after the 100 000th element, however version 3 has no such problem.

```BEGIN # finds elements of the sequence a(n) where a(n) is number of pairs    #
# (x,y) where 1 < x < y <= n that have at least one common prime       #
# factor. The sequence elements can be calculated by:                  #
# a(n) = n(n-1)/2 + 1 - sum i = 1..n of phi(i) where phi is Euler's    #
#                                                    totient function  #

MODE ELEMENT = LONG INT;   # integral type large enough for a(1 000 000) #

# returns the number of integers k where 1 <= k <= n that are mutually   #
#                                                             prime to n #
PROC totient = ( ELEMENT n )ELEMENT:         # algorithm from the second #
IF   n < 3 THEN 1           # Go Sample in the Totient function task #
ELIF n = 3 THEN 2
ELSE
ELEMENT result := n;
ELEMENT v      := n;
ELEMENT i      := 2;
WHILE i * i <= v DO
IF v MOD i = 0 THEN
WHILE v MOD i = 0 DO v OVERAB i OD;
result -:= result OVER i
FI;
IF i = 2 THEN
i := 1
FI;
i +:= 2
OD;
IF v > 1 THEN result -:= result OVER v FI;
result
FI # totient # ;

INT     max number    = 1 000 000;    # maximum number of terms required #
ELEMENT totient sum  := 0;                    # sum of the totients 1..n #
INT     next to show := 1 000;        # next power of 10 element to show #
ELEMENT n            := 0;
TO max number DO
n           +:= 1;
totient sum +:= totient( n );
ELEMENT an    = ( ( ( n * ( n - 1 ) ) OVER 2 ) + 1 ) - totient sum;
IF n <= 100 THEN
print( ( " ", whole( an, -4 ) ) );
IF n MOD 10 = 0 THEN print( ( newline ) ) FI
ELIF n = next to show THEN
print( ( "a(", whole( n, 0 ), "): ", whole( an, 0 ), newline ) );
next to show *:= 10
FI
OD
END```
Output:
```    0    0    0    1    1    4    4    7    9   14
14   21   21   28   34   41   41   52   52   63
71   82   82   97  101  114  122  137  137  158
158  173  185  202  212  235  235  254  268  291
291  320  320  343  363  386  386  417  423  452
470  497  497  532  546  577  597  626  626  669
669  700  726  757  773  818  818  853  877  922
922  969  969 1006 1040 1079 1095 1148 1148 1195
1221 1262 1262 1321 1341 1384 1414 1461 1461 1526
1544 1591 1623 1670 1692 1755 1755 1810 1848 1907
a(1000): 195309
a(10000): 19597515
a(100000): 1960299247
a(1000000): 196035947609
```

ALGOL W

As Algol W is limited to 32 bit integers, shows a(100000) but not a(1000000).

```begin % finds integers of the sequence a(n) where a(n) is number of pairs    %
% (x,y) where 1 < x < y <= n that have at least one common prime       %
% factor. The sequence integers can be calculated by:                  %
% a(n) = n(n-1)/2 + 1 - sum i = 1..n of phi(i) where phi is Euler's    %
%                                                    totient function  %

% returns the number of integers k where 1 <= k <= n that are mutually   %
%                                                             prime to n %
integer procedure totient ( integer value n ) ;     % algorithm from the %
if       n < 3 then 1   % 2nd Go Sample in the Totient function task %
else if  n = 3 then 2
else begin
integer t, v, i;
t := n;
v := n;
i := 2;
while i * i <= v do begin
if v rem i = 0 then begin
while v rem i = 0 do v := v div i;
t := t - t div i
end if_v_ren_i_eq_0 ;
if i = 2 then i := 1;
i := i + 2
end while_ii_le_v ;
if v > 1 then t - t div v else t
end totient ;

integer maxNumber, totientSum, nextToShow;
maxNumber  := 100000;                 % maximum number of terms required %
totientSum := 0;                              % sum of the totients 1..n %
nextToShow := 1000;                   % next power of 10 integer to show %
for n := 1 until maxNumber do begin
integer an;
totientSum := totientSum + totient( n );
an         := ( ( ( n * ( n - 1 ) ) div 2 ) + 1 ) - totientSum;
if n <= 100 then begin
writeon( i_w := 4, s_w := 0, " ", an );
if n rem 10 = 0 then write()
end
else if n = nextToShow then begin
write( i_w := 1, s_w := 0, "a(", n, "): ", an );
nextToShow := nextToShow * 10
end if_n_le_100__n_eq_nextToShow
end for_n

end.```
Output:
```    0    0    0    1    1    4    4    7    9   14
14   21   21   28   34   41   41   52   52   63
71   82   82   97  101  114  122  137  137  158
158  173  185  202  212  235  235  254  268  291
291  320  320  343  363  386  386  417  423  452
470  497  497  532  546  577  597  626  626  669
669  700  726  757  773  818  818  853  877  922
922  969  969 1006 1040 1079 1095 1148 1148 1195
1221 1262 1262 1321 1341 1384 1414 1461 1461 1526
1544 1591 1623 1670 1692 1755 1755 1810 1848 1907

a(1000): 195309
a(10000): 19597515
a(100000): 1960299247
```

C

Translation of: Wren
```#include <stdio.h>
#include <stdint.h>
#include <stdbool.h>
#include <stdlib.h>
#include <locale.h>

#define MAX 1000000

bool isPrime(int n) {
if (n < 2) return false;
if (n%2 == 0) return n == 2;
if (n%3 == 0) return n == 3;
int d = 5;
while (d*d <= n) {
if (n%d == 0) return false;
d += 2;
if (n%d == 0) return false;
d += 4;
}
return true;
}

uint64_t totient(uint64_t n) {
uint64_t tot = n, i = 2;
while (i*i <= n) {
if (!(n%i)) {
do {n /= i;} while (!(n%i));
tot -= tot/i;
}
if (i == 2) i = 1;
i += 2;
}
if (n > 1) tot -= tot/n;
}

const char *ord(int c) {
int m = c % 100;
if (m >= 4 && m <= 20) return "th";
m %= 10;
return (m == 1) ? "st" :
(m == 2) ? "nd" :
(m == 3) ? "rd" : "th";
}

int main() {
uint64_t n, sumPhi = 0, *a = (uint64_t *)calloc(MAX, sizeof(uint64_t));
int i, limit;
for (n = 1; n <= MAX; ++n) {
sumPhi += totient(n);
if (isPrime(n)) {
a[n-1] = a[n-2];
} else {
a[n-1] = n*(n-1)/2 + 1 - sumPhi;
}
}
setlocale(LC_NUMERIC, "");
printf("Number of pairs with common factors - first one hundred terms:\n");
for (i = 0; i < 100; ++i) {
printf("%'5lu  ", a[i]);
if (!((i+1)%10)) printf("\n");
}
printf("\n");
for (limit = 1; limit <= MAX; limit *= 10) {
printf("The %'d%s term: %'lu\n", limit, ord(limit), a[limit-1]);
}
free(a);
return 0;
}
```
Output:
```Number of pairs with common factors - first one hundred terms:
0      0      0      1      1      4      4      7      9     14
14     21     21     28     34     41     41     52     52     63
71     82     82     97    101    114    122    137    137    158
158    173    185    202    212    235    235    254    268    291
291    320    320    343    363    386    386    417    423    452
470    497    497    532    546    577    597    626    626    669
669    700    726    757    773    818    818    853    877    922
922    969    969  1,006  1,040  1,079  1,095  1,148  1,148  1,195
1,221  1,262  1,262  1,321  1,341  1,384  1,414  1,461  1,461  1,526
1,544  1,591  1,623  1,670  1,692  1,755  1,755  1,810  1,848  1,907

The 1st term: 0
The 10th term: 14
The 100th term: 1,907
The 1,000th term: 195,309
The 10,000th term: 19,597,515
The 100,000th term: 1,960,299,247
The 1,000,000th term: 196,035,947,609
```

C++

```#include <algorithm>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <numeric>
#include <vector>

std::vector<uint32_t> totients;
std::vector<uint32_t> primes;

void listTotients(const uint32_t& maximum) {
totients.resize(maximum + 1);
std::iota(totients.begin(), totients.end(), 0);

for ( uint32_t i = 2; i <= maximum; ++i ) {
if ( totients[i] == i ) {
totients[i] = i - 1;
for ( uint32_t j = i * 2; j <= maximum; j += i ) {
totients[j] = ( totients[j] / i ) * ( i - 1 );
}
}
}
}

const uint32_t halfMaximum = ( maximum + 1 ) / 2;
std::vector<bool> composite(halfMaximum, false);

for ( uint32_t i = 1, p = 3; i < halfMaximum; p += 2, ++i ) {
if ( ! composite[i] ) {
for ( uint32_t j = i + p; j < halfMaximum; j += p ) {
composite[j] = true;
}
}
}

primes.push_back(2);
for ( uint32_t i = 1, p = 3; i < halfMaximum; p += 2, ++i ) {
if ( ! composite[i] ) {
primes.push_back(p);
}
}
}

int main() {
const uint32_t maximum = 1'000'000;
listTotients(maximum);
std::vector<uint64_t> pairsCount(maximum + 1, 0);
uint64_t totientSum = 0;

for ( uint64_t number = 1; number <= maximum; ++number ) {
totientSum += totients[number];
if ( std::binary_search(primes.begin(), primes.end(), number) ) {
pairsCount[number] = pairsCount[number - 1];
} else {
pairsCount[number] = ( number * ( number - 1 ) >> 1 ) - totientSum + 1;
}
}

std::cout << "The first one hundred terms of the number of pairs with common factors:" << std::endl;
for ( uint32_t number = 1; number <= 100; ++number ) {
std::cout << std::setw(4) << pairsCount[number] << ( ( number % 10 == 0 ) ? "\n" : " " );
}
std::cout << std::endl;

uint32_t term = 1;
while ( term <= maximum ) {
std::cout << std::left << std::setw(14)
<< "Term " + std::to_string(term) + ": " << pairsCount[term] << std::endl;
term *= 10;
}
}
```
Output:
```The first one hundred terms of the number of pairs with common factors:
0    0    0    1    1    4    4    7    9   14
14   21   21   28   34   41   41   52   52   63
71   82   82   97  101  114  122  137  137  158
158  173  185  202  212  235  235  254  268  291
291  320  320  343  363  386  386  417  423  452
470  497  497  532  546  577  597  626  626  669
669  700  726  757  773  818  818  853  877  922
922  969  969 1006 1040 1079 1095 1148 1148 1195
1221 1262 1262 1321 1341 1384 1414 1461 1461 1526
1544 1591 1623 1670 1692 1755 1755 1810 1848 1907

Term 1:       0
Term 10:      14
Term 100:     1907
Term 1000:    195309
Term 10000:   19597515
Term 100000:  1960299247
Term 1000000: 196035947609
```

EasyLang

Translation of: C
```func isprim num .
if i < 2
return 0
.
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
func totient n .
tot = n
i = 2
while i * i <= n
if n mod i = 0
repeat
n /= i
until n mod i <> 0
.
tot -= tot / i
.
if i = 2
i = 1
.
i += 2
.
if n > 1
tot -= tot / n
.
.
write "1-100:"
for n = 1 to 1000
sumPhi += totient n
if isprim n = 1
a = ap
else
a = n * (n - 1) / 2 + 1 - sumPhi
.
if n <= 100
write " " & a
.
ap = a
.
print ""
print "1000: " & a```

Factor

Works with: Factor version 0.99 2022-04-03
```USING: formatting grouping io kernel math math.functions
math.primes.factors prettyprint ranges sequences
tools.memory.private ;

: totient-sum ( n -- sum )
[1..b] [ totient ] map-sum ;

: a ( n -- a(n) )
dup [ 1 - * 2 / ] keep totient-sum - ;

"Number of pairs with common factors - first 100 terms:" print
100 [1..b] [ a commas ] map 10 group simple-table. nl

7 <iota> [ dup 10^ a commas "Term #1e%d: %s\n" printf ] each
```
Output:
```Number of pairs with common factors - first 100 terms:
0     0     0     1     1     4     4     7     9     14
14    21    21    28    34    41    41    52    52    63
71    82    82    97    101   114   122   137   137   158
158   173   185   202   212   235   235   254   268   291
291   320   320   343   363   386   386   417   423   452
470   497   497   532   546   577   597   626   626   669
669   700   726   757   773   818   818   853   877   922
922   969   969   1,006 1,040 1,079 1,095 1,148 1,148 1,195
1,221 1,262 1,262 1,321 1,341 1,384 1,414 1,461 1,461 1,526
1,544 1,591 1,623 1,670 1,692 1,755 1,755 1,810 1,848 1,907

Term #1e0: 0
Term #1e1: 14
Term #1e2: 1,907
Term #1e3: 195,309
Term #1e4: 19,597,515
Term #1e5: 1,960,299,247
Term #1e6: 196,035,947,609
```

FreeBASIC

```Function isPrime(n As Uinteger) As Boolean
If n Mod 2 = 0 Then Return false
For i As Uinteger = 3 To Int(Sqr(n))+1 Step 2
If n Mod i = 0 Then Return false
Next i
Return true
End Function

Function Totient(n As Uinteger) As Integer
Dim As Uinteger tot = n, i = 2
While i*i <= n
If n Mod i = 0 Then
Do
n /= i
Loop Until n Mod i <> 0
tot -= tot/i
End If
i += Iif(i = 2, 1, 2)
Wend
If n > 1 Then tot -= tot/n
End Function

Dim As Uinteger n, limit = 1e6, sumPhi = 0
Dim As Uinteger a(limit)
For n = 1 To limit
sumPhi += Totient(n)
a(n) = Iif(isPrime(n), a(n-1), n * (n - 1) / 2 + 1 - sumPhi)
Next n

Print "Number of pairs with common factors - first one hundred terms:"
Dim As Uinteger j, count = 0
For j = 1 To 100
count += 1
Print Using "  ##,###"; a(j);
If(count Mod 10) = 0 Then Print
Next j

Print !"\nThe 1st term: "; a(1)
Print "The 10th term: "; a(10)
Print "The 100th term: "; a(1e2)
Print "The 1,000th term: "; a(1e3)
Print "The 10,000th term: "; a(1e4)
Print "The 100,000th term: "; a(1e5)
Print "The 1,000,000th term: "; a(1e6)
Sleep```
Output:
```Number of pairs with common factors - first one hundred terms:
0       0       0       1       1       4       4       7       9      14
14      21      21      28      34      41      41      52      52      63
71      82      82      97     101     114     122     137     137     158
158     173     185     202     212     235     235     254     268     291
291     320     320     343     363     386     386     417     423     452
470     497     497     532     546     577     597     626     626     669
669     700     726     757     773     818     818     853     877     922
922     969     969   1,006   1,040   1,079   1,095   1,148   1,148   1,195
1,221   1,262   1,262   1,321   1,341   1,384   1,414   1,461   1,461   1,526
1,544   1,591   1,623   1,670   1,692   1,755   1,755   1,810   1,848   1,907

The 1st term: 0
The 10th term: 14
The 100th term: 1907
The 1,000th term: 195309
The 10,000th term: 19597515
The 100,000th term: 1960299247
The 1,000,000th term: 196035947609```

Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"rcu"
)

func totient(n uint64) uint64 {
tot := n
i := uint64(2)
for i*i <= n {
if n%i == 0 {
for n%i == 0 {
n /= i
}
tot -= tot / i
}
if i == 2 {
i = 1
}
i += 2
}
if n > 1 {
tot -= tot / n
}
}

func ord(c int) string {
m := c % 100
if m >= 4 && m <= 20 {
return "th"
}
m %= 10
switch m {
case 1:
return "st"
case 2:
return "md"
case 3:
return "rd"
default:
return "th"
}
}

func main() {
const max = 1_000_000
a := make([]uint64, max)
sumPhi := uint64(0)
for n := uint64(1); n <= uint64(max); n++ {
sumPhi += totient(n)
if rcu.IsPrime(n) {
a[n-1] = a[n-2]
} else {
a[n-1] = n*(n-1)/2 + 1 - sumPhi
}
}
fmt.Println("Number of pairs with common factors - first one hundred terms:")
rcu.PrintTable(a[:100], 10, 6, true)
fmt.Println()
for limit := 1; limit <= max; limit *= 10 {
fmt.Printf("The %s%s term: %s\n", rcu.Commatize(limit), ord(limit), rcu.Commatize(a[limit-1]))
}
}
```
Output:
```Number of pairs with common factors - first one hundred terms:
0      0      0      1      1      4      4      7      9     14
14     21     21     28     34     41     41     52     52     63
71     82     82     97    101    114    122    137    137    158
158    173    185    202    212    235    235    254    268    291
291    320    320    343    363    386    386    417    423    452
470    497    497    532    546    577    597    626    626    669
669    700    726    757    773    818    818    853    877    922
922    969    969  1,006  1,040  1,079  1,095  1,148  1,148  1,195
1,221  1,262  1,262  1,321  1,341  1,384  1,414  1,461  1,461  1,526
1,544  1,591  1,623  1,670  1,692  1,755  1,755  1,810  1,848  1,907

The 1st term: 0
The 10th term: 14
The 100th term: 1,907
The 1,000th term: 195,309
The 10,000th term: 19,597,515
The 100,000th term: 1,960,299,247
The 1,000,000th term: 196,035,947,609
```

J

For this task, because of the summation of euler totient values, it's more efficient to generate the sequence with a slightly different routine than we would use to compute a single value. Thus:

```   (1 _1r2 1r2&p. - +/\@:(5&p:)) 1+i.1e2
0 0 0 1 1 4 4 7 9 14 14 21 21 28 34 41 41 52 52 63 71 82 82 97 101 114 122 137 137 158 158 173 185 202 212 235 235 254 268 291 291 320 320 343 363 386 386 417 423 452 470 497 497 532 546 577 597 626 626 669 669 700 726 757 773 818 818 853 877 922 922 969 969 1006 1040 1079 1095 1148 1148 1195 1221 1262 1262 1321 1341 1384 1414 1461 1461 1526 1544 1591 1623 1670 1692 1755 1755 1810 1848 1907
(1 _1r2 1r2&p.@{: - +/@:(5&p:)) 1+i.1e3
195309
(1 _1r2 1r2&p.@{: - +/@:(5&p:)) 1+i.1e4
19597515
(1 _1r2 1r2&p.@{: - +/@:(5&p:)) 1+i.1e5
1960299247
(1 _1r2 1r2&p.@{: - +/@:(5&p:)) 1+i.1e6
196035947609
```

Here, `p.` calculates a polynomial (1 + (-x)/2 + (x^2)/2 in this example), `5&p:` is euler's totient function, `@{:` modifies the polynomial to only operate on the final element of a sequence, `+/` is sum and `+/\` is running sum, and `1+i.n` is the sequence of numbers 1 through n.

Java

```import java.util.ArrayList;
import java.util.Collections;
import java.util.List;

public final class PairsWithCommonFactors {

public static void main(String[] args) {
final int maximum = 1_000_000;
listTotients(maximum);
long[] pairsCount = new long[maximum + 1];
long totientSum = 0;

for ( int number = 1; number <= maximum; number++ ) {
totientSum += totients[number];
if ( Collections.binarySearch(primes, number) > 0 ) {
pairsCount[number] = pairsCount[number - 1];
} else {
pairsCount[number] = ( (long) number * ( number - 1 ) >> 1 ) - totientSum + 1;
}
}

System.out.println("The first one hundred terms of the number of pairs with common factors:");
for ( int number = 1; number <= 100; number++ ) {
System.out.print(String.format("%4d%s", pairsCount[number], ( ( number % 10 == 0 ) ? "\n" : " " )));
}
System.out.println();

int term = 1;
while ( term <= maximum ) {
System.out.println(String.format("%-14s%s", "Term " + term + ": ", pairsCount[term]));
term *= 10;
}
}

private static void listTotients(int maximum) {
totients = new int[maximum + 1];
for ( int i = 0; i <= maximum; i++ ) {
totients[i] = i;
}

for ( int i = 2; i <= maximum; i++ ) {
if ( totients[i] == i ) {
totients[i] = i - 1;
for ( int j = i * 2; j <= maximum; j += i ) {
totients[j] = ( totients[j] / i ) * ( i - 1 );
}
}
}
}

private static void listPrimeNumbers(int maximum) {
final int halfMaximum = ( maximum + 1 ) / 2;
boolean[] composite = new boolean[halfMaximum];
for ( int i = 1, p = 3; i < halfMaximum; p += 2, i++ ) {
if ( ! composite[i] ) {
for ( int j = i + p; j < halfMaximum; j += p ) {
composite[j] = true;
}
}
}

primes = new ArrayList<Integer>(List.of( 2 ));
for ( int i = 1, p = 3; i < halfMaximum; p += 2, i++ ) {
if ( ! composite[i] ) {
}
}
}

private static int[] totients;
private static List<Integer> primes;

}
```
Output:
```The first one hundred terms of the number of pairs with common factors:
0    0    0    1    1    4    4    7    9   14
14   21   21   28   34   41   41   52   52   63
71   82   82   97  101  114  122  137  137  158
158  173  185  202  212  235  235  254  268  291
291  320  320  343  363  386  386  417  423  452
470  497  497  532  546  577  597  626  626  669
669  700  726  757  773  818  818  853  877  922
922  969  969 1006 1040 1079 1095 1148 1148 1195
1221 1262 1262 1321 1341 1384 1414 1461 1461 1526
1544 1591 1623 1670 1692 1755 1755 1810 1848 1907

Term 1:       0
Term 10:      14
Term 100:     1907
Term 1000:    195309
Term 10000:   19597515
Term 100000:  1960299247
Term 1000000: 196035947609
```

jq

Works with jq and gojq, that is, the C and Go implementations of jq.

If using jq, the definition of `_nwise` can be omitted.

Preliminaries

```# For the sake of gojq
def _nwise(\$n):
def n: if length <= \$n then . else .[0:\$n] , (.[\$n:] | n) end;
n;

def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l)[:\$l] + .;

def is_prime:
. as \$n
| if (\$n < 2)         then false
elif (\$n % 2 == 0)  then \$n == 2
elif (\$n % 3 == 0)  then \$n == 3
elif (\$n % 5 == 0)  then \$n == 5
elif (\$n % 7 == 0)  then \$n == 7
elif (\$n % 11 == 0) then \$n == 11
elif (\$n % 13 == 0) then \$n == 13
elif (\$n % 17 == 0) then \$n == 17
elif (\$n % 19 == 0) then \$n == 19
else
(\$n | sqrt) as \$rt
| 23
| until( . > \$rt or (\$n % . == 0); .+2)
| . > \$rt
end;

# jq optimizes the recursive call of _gcd in the following:
def gcd(a;b):
def _gcd:
if .[1] != 0 then [.[1], .[0] % .[1]] | _gcd else .[0] end;
[a,b] | _gcd ;

def count(s): reduce s as \$x (0; .+1);

def totient:
. as \$n
| count( range(0; .) | select( gcd(\$n; .) == 1) );```

```def sumPhi(\$max):
reduce range(1; max+1) as \$n ({};
.sumPhi += (\$n|totient)
| if \$n | is_prime
then .a[\$n-1] = .a[\$n-2]
else
.a[\$n-1] = \$n * (\$n - 1) / 2 + 1 - .sumPhi
end ) ;

def limits: [ 1, 10, 1e2, 1e3 ];

"Number of pairs with common factors - first one hundred terms:",
(sumPhi( limits[-1] )
| (.a[0:100] | _nwise(10) | map(lpad(6)) | join(" ") ),
( limits[] as \$i
| "The #\(\$i) term: \(.a[\$i-1])" ) )```
Output:
```     0      0      0      1      1      4      4      7      9     14
14     21     21     28     34     41     41     52     52     63
71     82     82     97    101    114    122    137    137    158
158    173    185    202    212    235    235    254    268    291
291    320    320    343    363    386    386    417    423    452
470    497    497    532    546    577    597    626    626    669
669    700    726    757    773    818    818    853    877    922
922    969    969   1006   1040   1079   1095   1148   1148   1195
1221   1262   1262   1321   1341   1384   1414   1461   1461   1526
1544   1591   1623   1670   1692   1755   1755   1810   1848   1907
The #1 term: 0
The #10 term: 14
The #100 term: 1907
The #1000 term: 195309
```

Julia

```using Formatting
using Primes

pcf(n) = n * (n - 1) ÷ 2 + 1 - sum(totient, 1:n)

foreach(p -> print(rpad(p[2], 5), p[1] % 20 == 0 ? "\n" : ""), pairs(map(pcf, 1:100)))

for expo in 1:6
println("The ", format(10^expo, commas = true), "th pair with common factors count is ",
format(pcf(10^expo), commas = true))
end
```
Output:
```0    0    0    1    1    4    4    7    9    14   14   21   21   28   34   41   41   52   52   63
71   82   82   97   101  114  122  137  137  158  158  173  185  202  212  235  235  254  268  291
291  320  320  343  363  386  386  417  423  452  470  497  497  532  546  577  597  626  626  669
669  700  726  757  773  818  818  853  877  922  922  969  969  1006 1040 1079 1095 1148 1148 1195
1221 1262 1262 1321 1341 1384 1414 1461 1461 1526 1544 1591 1623 1670 1692 1755 1755 1810 1848 1907
The 10th pair with common factors count is 14
The 100th pair with common factors count is 1,907
The 1,000th pair with common factors count is 195,309
The 10,000th pair with common factors count is 19,597,515
The 100,000th pair with common factors count is 1,960,299,247
The 1,000,000th pair with common factors count is 196,035,947,609
```

Mathematica / Wolfram Language

Translation of: Julia
```(* Define the prime counting function (pcf) *)
pcf[n_] := n (n - 1)/2 + 1 - Total[EulerPhi /@ Range[n]]

(* Print pairs of (n, pcf(n)) for n from 1 to 100 *)
Do[
Module[{pair = pcf[n], pairString},
If[Mod[n, 20] == 0,
WriteString["stdout", pairString <> "\n"],
WriteString["stdout", pairString, " "]
]
],
{n, 1, 100}
]

(* Print the 10^expo-th pair with common factors count *)
Do[
Print["The ", ToString[NumberForm[10^expo, DigitBlock -> 3]],
"th pair with common factors count is ",
ToString[NumberForm[pcf[10^expo], DigitBlock -> 3]]],
{expo, 1, 6}
]
```
Output:
```0     0     0     1     1     4     4     7     9     14    14    21    21    28    34    41    41    52    52    63
71    82    82    97    101   114   122   137   137   158   158   173   185   202   212   235   235   254   268   291
291   320   320   343   363   386   386   417   423   452   470   497   497   532   546   577   597   626   626   669
669   700   726   757   773   818   818   853   877   922   922   969   969   1006  1040  1079  1095  1148  1148  1195
1221  1262  1262  1321  1341  1384  1414  1461  1461  1526  1544  1591  1623  1670  1692  1755  1755  1810  1848  1907
The 10th pair with common factors count is 14
The 100th pair with common factors count is 1,907
The 1,000th pair with common factors count is 195,309
The 10,000th pair with common factors count is 19,597,515
The 100,000th pair with common factors count is 1,960,299,247
The 1,000,000th pair with common factors count is 196,035,947,609
```

Maxima

```/* Define the prime counting function (pcf) */
pcf(n) := n*(n - 1)/2 + 1 - sum(totient(i), i, 1, n);

/* Print pairs of (n, pcf(n)) for n from 1 to 100 */
for n:1 thru 100 do (
pcf_n : ev(pcf(n), numer),
if mod(n, 20) = 0 then (
printf(true, "~4d~%", pcf_n)
) else (
printf(true, "~4d ", pcf_n)
)
);

/* Print the 10^expo-th pair with common factors count */
for expo:1 thru 6 do (
pcf_10expo : ev(pcf(10^expo), numer),
printf(true, "The ~a-th pair with common factors count is ~a~%", 10^expo, pcf_10expo)
);
```
Output:
```   0    0    0    1    1    4    4    7    9   14   14   21   21   28   34   41   41   52   52   63
71   82   82   97  101  114  122  137  137  158  158  173  185  202  212  235  235  254  268  291
291  320  320  343  363  386  386  417  423  452  470  497  497  532  546  577  597  626  626  669
669  700  726  757  773  818  818  853  877  922  922  969  969 1006 1040 1079 1095 1148 1148 1195
1221 1262 1262 1321 1341 1384 1414 1461 1461 1526 1544 1591 1623 1670 1692 1755 1755 1810 1848 1907
The 10-th pair with common factors count is 14
The 100-th pair with common factors count is 1907
The 1000-th pair with common factors count is 195309
The 10000-th pair with common factors count is 19597515
The 100000-th pair with common factors count is 1960299247
The 1000000-th pair with common factors count is 196035947609
```

Nim

Translation of: Wren
```import std/[sequtils, strutils]

func isPrime(n: Natural): bool =
if n < 2: return false
if (n and 1) == 0: return n == 2
if n mod 3 == 0: return n == 3
var k = 5
while k * k <= n:
if n mod k == 0 or n mod (k + 2) == 0: return false
inc k, 6
result = true

func totient(n: Natural): int =
var n = n
result = n
var i = 2
while i * i <= n:
if n mod i == 0:
while n mod i == 0:
n = n div i
dec result, result div i
if i == 2: i = 1
inc i, 2
if n > 1:
dec result, result div n

const Max = 1_000_000
var a: array[Max, int]
var sumPhi = 0
for n in 1..Max:
inc sumPhi, n.totient
if n.isPrime:
a[n - 1] = a[n - 2]
else:
a[n-1] = n * (n - 1) shr 1 + 1 - sumPhi

echo "Number of pairs with common factors - First one hundred terms:"
for n in countup(0, 99, 10):
echo a[n..n+9].mapIt(align(\$it, 4)).join(" ")
echo()
var limit = 1
while limit <= Max:
echo "The \$1th term: \$2".format(insertSep(\$limit), insertSep(\$a[limit-1]))
limit *= 10
```
Output:
```Number of pairs with common factors - First one hundred terms:
0    0    0    1    1    4    4    7    9   14
14   21   21   28   34   41   41   52   52   63
71   82   82   97  101  114  122  137  137  158
158  173  185  202  212  235  235  254  268  291
291  320  320  343  363  386  386  417  423  452
470  497  497  532  546  577  597  626  626  669
669  700  726  757  773  818  818  853  877  922
922  969  969 1006 1040 1079 1095 1148 1148 1195
1221 1262 1262 1321 1341 1384 1414 1461 1461 1526
1544 1591 1623 1670 1692 1755 1755 1810 1848 1907

The 1th term: 0
The 10th term: 14
The 100th term: 1_907
The 1_000th term: 195_309
The 10_000th term: 19_597_515
The 100_000th term: 1_960_299_247
The 1_000_000th term: 196_035_947_609
```

Pascal

Free Pascal

modified Perfect_totient_numbers

```program PairsWithCommonFactors;
{\$IFdef FPC} {\$MODE DELPHI} {\$Optimization ON,ALL}{\$IFEND}
{\$IFdef Windows} {\$APPTYPE CONSOLE}{\$IFEND}
const
cLimit = 1000*1000*1000;
//global
type
TElem= Uint64;
tpElem = pUint64;

myString = String[31];

var
TotientList : array of TElem;
Sieve : Array of byte;

function Numb2USA(n:Uint64):myString;
const
//extend s by the count of comma to be inserted
deltaLength : array[0..24] of byte =
(0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7);
var
pI :pChar;
i,j : NativeInt;
Begin
str(n,result);
i := length(result);
//extend s by the count of comma to be inserted
// j := i+ (i-1) div 3;
j := i+deltaLength[i];
if i<> j then
Begin
setlength(result,j);
pI := @result[1];
dec(pI);
while i > 3 do
Begin
//copy 3 digits
pI[j] := pI[i];
pI[j-1] := pI[i-1];
pI[j-2] := pI[i-2];
// insert comma
pI[j-3] := ',';
dec(i,3);
dec(j,4);
end;
end;
end;

procedure SieveInit(svLimit:NativeUint);
var
pSieve:pByte;
i,j,pr :NativeUint;
Begin
svlimit := (svLimit+1) DIV 2;
setlength(sieve,svlimit+1);
pSieve := @Sieve[0];
For i := 1 to svlimit do
Begin
IF pSieve[i]= 0 then
Begin
pr := 2*i+1;
j := (sqr(pr)-1) DIV 2;
IF  j> svlimit then
BREAK;
repeat
pSieve[j]:= 1;
inc(j,pr);
until j> svlimit;
end;
end;
pr := 0;
j := 0;
For i := 1 to svlimit do
Begin
IF pSieve[i]= 0 then
Begin
pSieve[j] := i-pr;
inc(j);
pr := i;
end;
end;
setlength(sieve,j);
end;

procedure TotientInit(len: NativeUint);
var
pTotLst : tpElem;
pSieve  : pByte;
i: NativeInt;
p,j,k,svLimit : NativeUint;
Begin
SieveInit(len);
setlength(TotientList,len+12);
pTotLst := @TotientList[0];

//Fill totient with simple start values for odd and even numbers
//and multiples of 3
j := 1;
k := 1;// k == j DIV 2
p := 1;// p == j div 3;
repeat
pTotLst[j] := j;//1
pTotLst[j+1] := k;//2 j DIV 2; //2
inc(k);
inc(j,2);
pTotLst[j] := j-p;//3
inc(p);
pTotLst[j+1] := k;//4  j div 2
inc(k);
inc(j,2);
pTotLst[j] := j;//5
pTotLst[j+1] := p;//6   j DIV 3 <=  (div 2) * 2 DIV/3
inc(j,2);
inc(p);
inc(k);
until j>len+6;

//correct values of totient by prime factors
svLimit := High(sieve);
p := 3;// starting after 3
pSieve := @Sieve[svLimit+1];
i := -svlimit;
repeat
p := p+2*pSieve[i];
j := p;
while j <= cLimit do
Begin
k:= pTotLst[j];
pTotLst[j]:= k-(k DIV p);
inc(j,p);
end;
inc(i);
until i=0;
//primes not needed anymore
setlength(sieve,0);
end;

procedure CountOfPairs(len: NativeUint);
var
pTotLst : tpElem;
i,a_n,sum,Totient: tElem;
Begin
pTotLst := @TotientList[0];
sum := 1;
a_n := 2; // sums to i*(i-1)/2 +1
For i := 2 to len do
Begin
Totient := pTotLst[i];// relict for print data
sum += Totient;
pTotLst[i] := a_n-sum;
a_n += i;
end;
TotientList[1] := 0;
end;

var
i,k : NativeUint;
Begin
TotientInit(climit);
CountOfPairs(climit);
i := 1;
Repeat
For k := 9 downto 0 do
begin
write(TotientList[i]:6);
inc(i);
end;
writeln;
until i>99;
writeln;
writeln('Some values #');
i := 10;
repeat
writeln(Numb2USA(i):13,Numb2USA(TotientList[i]):25);
i *= 10;
until i > climit;
end.
```
Output:
```     0     0     0     1     1     4     4     7     9    14
14    21    21    28    34    41    41    52    52    63
71    82    82    97   101   114   122   137   137   158
158   173   185   202   212   235   235   254   268   291
291   320   320   343   363   386   386   417   423   452
470   497   497   532   546   577   597   626   626   669
669   700   726   757   773   818   818   853   877   922
922   969   969  1006  1040  1079  1095  1148  1148  1195
1221  1262  1262  1321  1341  1384  1414  1461  1461  1526
1544  1591  1623  1670  1692  1755  1755  1810  1848  1907

Some values #
10                       14
100                    1,907
1,000                  195,309
10,000               19,597,515
100,000            1,960,299,247
1,000,000          196,035,947,609
10,000,000       19,603,638,572,759
100,000,000    1,960,364,433,634,093
1,000,000,000  196,036,448,326,991,587
real    0m23,577s```

Perl

Library: ntheory
```use v5.36;
use ntheory 'factor';
use List::Util qw<sum product uniq max>;

sub comma { reverse ((reverse shift) =~ s/(.{3})/\$1,/gr) =~ s/^,//r }
sub table (\$c, @V) { my \$t = \$c * (my \$w = 2 + max map {length} @V); ( sprintf( ('%'.\$w.'s')x@V, @V) ) =~ s/.{1,\$t}\K/\n/gr }

my(\$max, @phi, @n_pairs) = (100, 0);
for my \$t (1..\$max) { push @phi, \$t * product map { 1 - 1/\$_ } uniq factor(\$t) }
push @n_pairs, comma \$_ * (\$_ - 1) / 2 + 1 - sum @phi[1..\$_] for 1..\$max;

say 'Number of pairs with common factors - first one hundred terms:';
say table 10, @n_pairs;
```
Output:
```Number of pairs with common factors - first one hundred terms:
0      0      0      1      1      4      4      7      9     14
14     21     21     28     34     41     41     52     52     63
71     82     82     97    101    114    122    137    137    158
158    173    185    202    212    235    235    254    268    291
291    320    320    343    363    386    386    417    423    452
470    497    497    532    546    577    597    626    626    669
669    700    726    757    773    818    818    853    877    922
922    969    969  1,006  1,040  1,079  1,095  1,148  1,148  1,195
1,221  1,262  1,262  1,321  1,341  1,384  1,414  1,461  1,461  1,526
1,544  1,591  1,623  1,670  1,692  1,755  1,755  1,810  1,848  1,907```

Phix

```with javascript_semantics
constant limit = 1e6
sequence a = repeat(0,limit)
atom sumPhi = 0
for n=1 to limit do
sumPhi += phi(n)
if is_prime(n) then
a[n] = a[n-1]
else
a[n] = n * (n - 1) / 2 + 1 - sumPhi
end if
end for

string j = join_by(a[1..100],1,10,fmt:="%,5d")
printf(1,"Number of pairs with common factors - first one hundred terms:\n%s\n",j)
for l in {1, 10, 1e2, 1e3, 1e4, 1e5, 1e6} do
printf(1,"%22s term: %,d\n", {proper(ordinal(l),"SENTENCE"), a[l]})
end for
```
Output:
```Number of pairs with common factors - first one hundred terms:
0       0       0       1       1       4       4       7       9      14
14      21      21      28      34      41      41      52      52      63
71      82      82      97     101     114     122     137     137     158
158     173     185     202     212     235     235     254     268     291
291     320     320     343     363     386     386     417     423     452
470     497     497     532     546     577     597     626     626     669
669     700     726     757     773     818     818     853     877     922
922     969     969   1,006   1,040   1,079   1,095   1,148   1,148   1,195
1,221   1,262   1,262   1,321   1,341   1,384   1,414   1,461   1,461   1,526
1,544   1,591   1,623   1,670   1,692   1,755   1,755   1,810   1,848   1,907

First term: 0
Tenth term: 14
One hundredth term: 1,907
One thousandth term: 195,309
Ten thousandth term: 19,597,515
One hundred thousandth term: 1,960,299,247
One millionth term: 196,035,947,609
```

Quackery

`totient` is defined at Totient function#Quackery.

```  [] 0
1000 times
[ i^ 1+ totient +
i^ 1+ dup 1 - * 2 / 1+
over -
swap dip join ]
drop
dup -1 peek swap
100 split drop
say "First 100 terms:"
[] swap witheach
[ number\$ nested join ]
60 wrap\$
cr cr say "1000th term: " echo```
Output:
```First 100 terms:
0 0 0 1 1 4 4 7 9 14 14 21 21 28 34 41 41 52 52 63 71 82 82
97 101 114 122 137 137 158 158 173 185 202 212 235 235 254
268 291 291 320 320 343 363 386 386 417 423 452 470 497 497
532 546 577 597 626 626 669 669 700 726 757 773 818 818 853
877 922 922 969 969 1006 1040 1079 1095 1148 1148 1195 1221
1262 1262 1321 1341 1384 1414 1461 1461 1526 1544 1591 1623
1670 1692 1755 1755 1810 1848 1907

1000th term: 195309```

Raku

```use Prime::Factor;
use Lingua::EN::Numbers;

my \𝜑 = 0, |(1..*).hyper.map: -> \t { t × [×] t.&prime-factors.unique.map: { 1 - 1/\$_ } }

sub pair-count (\n) {  n × (n - 1) / 2 + 1 - sum 𝜑[1..n] }

say "Number of pairs with common factors - first one hundred terms:\n"
~ (1..100).map(&pair-count).batch(10)».&comma».fmt("%6s").join("\n") ~ "\n";

for ^7 { say (my \$i = 10 ** \$_).&ordinal.tc.fmt("%22s term: ") ~ \$i.&pair-count.&comma }
```
Output:
```Number of pairs with common factors - first one hundred terms:
0      0      0      1      1      4      4      7      9     14
14     21     21     28     34     41     41     52     52     63
71     82     82     97    101    114    122    137    137    158
158    173    185    202    212    235    235    254    268    291
291    320    320    343    363    386    386    417    423    452
470    497    497    532    546    577    597    626    626    669
669    700    726    757    773    818    818    853    877    922
922    969    969  1,006  1,040  1,079  1,095  1,148  1,148  1,195
1,221  1,262  1,262  1,321  1,341  1,384  1,414  1,461  1,461  1,526
1,544  1,591  1,623  1,670  1,692  1,755  1,755  1,810  1,848  1,907

First term: 0
Tenth term: 14
One hundredth term: 1,907
One thousandth term: 195,309
Ten thousandth term: 19,597,515
One hundred thousandth term: 1,960,299,247
One millionth term: 196,035,947,609```

RPL

The `PHI` function is defined at Totient function To save time, Σφ(j) are stored in a global variable named `ΣPHI`, which must be initialized at `{ 1 }` once.

Translation of: C
Works with: Halcyon Calc version 4.2.8
RPL code Comment
```≪ ∑PHI SIZE
IF DUP2 ≤ THEN DROP
ELSE
'∑PHI' OVER GET SWAP 1 + ∑PHI SWAP 4 ROLL FOR j
SWAP j PHI + SWAP OVER + NEXT
'∑PHI' STO DROP ∑PHI SIZE END
'∑PHI' SWAP GET
≫ '→∑PHI' STO

≪ DUP DUP 1 - * 2 / 1 + SWAP →∑PHI - ≫ 'A185670' STO
```
```'→∑PHI' ( n -- φ(n) )
if n...
initialize stack and loop, for size(∑PHI)+1 to n
sum += φ(n), append sum to list
store list into ∑PHI

'A185670' ( n -- n*(n-1)/2 + 1 - Σφ(j) )
```
```≪ { } 1 100 FOR j j A185670 + NEXT ≫ EVAL
1000 A185670
```
Works with: HP version 49
```« DUP DUP 1 - 2 * / 1 +
'j' 4 ROLL 'EULER(j)' ∑ - R→I
» 'A185670' STO

« « n A185670 » 1 100 1 SEQ
1000 A185670
```
Output:
```2: { 0 0 0 1 1 4 4 7 9 14 14 21 21 28 34 41 41 52 52 63 71 82 82 97 101 114 122 137 137 158 158 173 185 202 212 235 235 254 268 291 291 320 320 343 363 386 386 417 423 452 470 497 497 532 546 577 597 626 626 669 669 700 726 757 773 818 818 853 877 922 922 969 969 1006 1040 1079 1095 1148 1148 1195 1221 1262 1262 1321 1341 1384 1414 1461 1461 1526 1544 1591 1623 1670 1692 1755 1755 1810 1848 1907 }
1: 195309
```

Ruby

```require "prime"

def 𝜑(n) = n.prime_division.inject(1) {|res, (pr, exp)| res *= (pr-1) * pr**(exp-1) }
def a(n) = n*(n-1)/2 + 1 - (1..n).sum{|i| 𝜑(i)}

puts "Number of pairs with common factors - first 100 terms: "
puts (1..100).map{|n| a(n) }.join(", ")
(1..6).each{|e| puts "Term #1e#{e}: #{a(10**e)}"}
```
Output:
```Number of pairs with common factors - first 100 terms:
0, 0, 0, 1, 1, 4, 4, 7, 9, 14, 14, 21, 21, 28, 34, 41, 41, 52, 52, 63, 71, 82, 82, 97, 101, 114, 122, 137, 137, 158, 158, 173, 185, 202, 212, 235, 235, 254, 268, 291, 291, 320, 320, 343, 363, 386, 386, 417, 423, 452, 470, 497, 497, 532, 546, 577, 597, 626, 626, 669, 669, 700, 726, 757, 773, 818, 818, 853, 877, 922, 922, 969, 969, 1006, 1040, 1079, 1095, 1148, 1148, 1195, 1221, 1262, 1262, 1321, 1341, 1384, 1414, 1461, 1461, 1526, 1544, 1591, 1623, 1670, 1692, 1755, 1755, 1810, 1848, 1907
1e1: 14
1e2: 1907
1e3: 195309
1e4: 19597515
1e5: 1960299247
1e6: 196035947609
```

Wren

Library: Wren-math
Library: Wren-fmt
```import "./math" for Int
import "./fmt" for Fmt

var totient = Fn.new { |n|
var tot = n
var i = 2
while (i*i <= n) {
if (n%i == 0) {
while(n%i == 0) n = (n/i).floor
tot = tot - (tot/i).floor
}
if (i == 2) i = 1
i = i + 2
}
if (n > 1) tot = tot - (tot/n).floor
}

var max = 1e6
var a = List.filled(max, 0)
var sumPhi = 0
for (n in 1..max) {
sumPhi = sumPhi + totient.call(n)
if (Int.isPrime(n)) {
a[n-1] = a[n-2]
} else {
a[n-1] = n * (n - 1) / 2 + 1 - sumPhi
}
}

System.print("Number of pairs with common factors - first one hundred terms:")
Fmt.tprint("\$,5d ", a[0..99], 10)
System.print()
var limits = [1, 10, 1e2, 1e3, 1e4, 1e5, 1e6]
for (limit in limits) {
Fmt.print("The \$,r term: \$,d", limit, a[limit-1])
}
```
Output:
```Number of pairs with common factors - first one hundred terms:
0      0      0      1      1      4      4      7      9     14
14     21     21     28     34     41     41     52     52     63
71     82     82     97    101    114    122    137    137    158
158    173    185    202    212    235    235    254    268    291
291    320    320    343    363    386    386    417    423    452
470    497    497    532    546    577    597    626    626    669
669    700    726    757    773    818    818    853    877    922
922    969    969  1,006  1,040  1,079  1,095  1,148  1,148  1,195
1,221  1,262  1,262  1,321  1,341  1,384  1,414  1,461  1,461  1,526
1,544  1,591  1,623  1,670  1,692  1,755  1,755  1,810  1,848  1,907

The 1st term: 0
The 10th term: 14
The 100th term: 1,907
The 1,000th term: 195,309
The 10,000th term: 19,597,515
The 100,000th term: 1,960,299,247
The 1,000,000th term: 196,035,947,609
```