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Sequence: nth number with exactly n divisors

From Rosetta Code
Task
Sequence: nth number with exactly n divisors
You are encouraged to solve this task according to the task description, using any language you may know.

Calculate the sequence where each term an is the nth that has n divisors.

Task

Show here, on this page, at least the first 15 terms of the sequence.

See also
Related tasks

C[edit]

Translation of: C++
#include <math.h>
#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>
 
#define LIMIT 15
int smallPrimes[LIMIT];
 
static void sieve() {
int i = 2, j;
int p = 5;
 
smallPrimes[0] = 2;
smallPrimes[1] = 3;
 
while (i < LIMIT) {
for (j = 0; j < i; j++) {
if (smallPrimes[j] * smallPrimes[j] <= p) {
if (p % smallPrimes[j] == 0) {
p += 2;
break;
}
} else {
smallPrimes[i++] = p;
p += 2;
break;
}
}
}
}
 
static bool is_prime(uint64_t n) {
uint64_t i;
 
for (i = 0; i < LIMIT; i++) {
if (n % smallPrimes[i] == 0) {
return n == smallPrimes[i];
}
}
 
i = smallPrimes[LIMIT - 1] + 2;
for (; i * i <= n; i += 2) {
if (n % i == 0) {
return false;
}
}
 
return true;
}
 
static uint64_t divisor_count(uint64_t n) {
uint64_t count = 1;
uint64_t d;
 
while (n % 2 == 0) {
n /= 2;
count++;
}
 
for (d = 3; d * d <= n; d += 2) {
uint64_t q = n / d;
uint64_t r = n % d;
uint64_t dc = 0;
while (r == 0) {
dc += count;
n = q;
q = n / d;
r = n % d;
}
count += dc;
}
 
if (n != 1) {
return count *= 2;
}
return count;
}
 
static uint64_t OEISA073916(size_t n) {
uint64_t count = 0;
uint64_t result = 0;
size_t i;
 
if (is_prime(n)) {
return (uint64_t)pow(smallPrimes[n - 1], n - 1);
}
 
for (i = 1; count < n; i++) {
if (n % 2 == 1) {
// The solution for an odd (non-prime) term is always a square number
uint64_t root = (uint64_t)sqrt(i);
if (root * root != i) {
continue;
}
}
if (divisor_count(i) == n) {
count++;
result = i;
}
}
 
return result;
}
 
int main() {
size_t n;
 
sieve();
 
for (n = 1; n <= LIMIT; n++) {
if (n == 13) {
printf("A073916(%lu) = One more bit needed to represent result.\n", n);
} else {
printf("A073916(%lu) = %llu\n", n, OEISA073916(n));
}
}
 
return 0;
}
Output:
A073916(1) = 1
A073916(2) = 3
A073916(3) = 25
A073916(4) = 14
A073916(5) = 14641
A073916(6) = 44
A073916(7) = 24137569
A073916(8) = 70
A073916(9) = 1089
A073916(10) = 405
A073916(11) = 819628286980801
A073916(12) = 160
A073916(13) = One more bit needed to represent result.
A073916(14) = 2752
A073916(15) = 9801

C++[edit]

Translation of: Java
#include <iostream>
#include <vector>
 
std::vector<int> smallPrimes;
 
bool is_prime(size_t test) {
if (test < 2) {
return false;
}
if (test % 2 == 0) {
return test == 2;
}
for (size_t d = 3; d * d <= test; d += 2) {
if (test % d == 0) {
return false;
}
}
return true;
}
 
void init_small_primes(size_t numPrimes) {
smallPrimes.push_back(2);
 
int count = 0;
for (size_t n = 3; count < numPrimes; n += 2) {
if (is_prime(n)) {
smallPrimes.push_back(n);
count++;
}
}
}
 
size_t divisor_count(size_t n) {
size_t count = 1;
while (n % 2 == 0) {
n /= 2;
count++;
}
for (size_t d = 3; d * d <= n; d += 2) {
size_t q = n / d;
size_t r = n % d;
size_t dc = 0;
while (r == 0) {
dc += count;
n = q;
q = n / d;
r = n % d;
}
count += dc;
}
if (n != 1) {
count *= 2;
}
return count;
}
 
uint64_t OEISA073916(size_t n) {
if (is_prime(n)) {
return (uint64_t) pow(smallPrimes[n - 1], n - 1);
}
 
size_t count = 0;
uint64_t result = 0;
for (size_t i = 1; count < n; i++) {
if (n % 2 == 1) {
// The solution for an odd (non-prime) term is always a square number
size_t root = (size_t) sqrt(i);
if (root * root != i) {
continue;
}
}
if (divisor_count(i) == n) {
count++;
result = i;
}
}
return result;
}
 
int main() {
const int MAX = 15;
init_small_primes(MAX);
for (size_t n = 1; n <= MAX; n++) {
if (n == 13) {
std::cout << "A073916(" << n << ") = One more bit needed to represent result.\n";
} else {
std::cout << "A073916(" << n << ") = " << OEISA073916(n) << '\n';
}
}
 
return 0;
}
Output:
A073916(1) = 1
A073916(2) = 3
A073916(3) = 25
A073916(4) = 14
A073916(5) = 14641
A073916(6) = 44
A073916(7) = 24137569
A073916(8) = 70
A073916(9) = 1089
A073916(10) = 405
A073916(11) = 819628286980801
A073916(12) = 160
A073916(13) = One more bit needed to represent result.
A073916(14) = 2752
A073916(15) = 9801

D[edit]

Translation of: Java
import std.bigint;
import std.math;
import std.stdio;
 
bool isPrime(long test) {
if (test == 2) {
return true;
}
if (test % 2 == 0) {
return false;
}
for (long d = 3 ; d * d <= test; d += 2) {
if (test % d == 0) {
return false;
}
}
return true;
}
 
int[] calcSmallPrimes(int numPrimes) {
int[] smallPrimes;
smallPrimes ~= 2;
 
int count = 0;
int n = 3;
while (count < numPrimes) {
if (isPrime(n)) {
smallPrimes ~= n;
count++;
}
n += 2;
}
 
return smallPrimes;
}
 
immutable MAX = 45;
immutable smallPrimes = calcSmallPrimes(MAX);
 
int getDivisorCount(long n) {
int count = 1;
while (n % 2 == 0) {
n /= 2;
count += 1;
}
for (long d = 3; d * d <= n; d += 2) {
long q = n / d;
long r = n % d;
int dc = 0;
while (r == 0) {
dc += count;
n = q;
q = n / d;
r = n % d;
}
count += dc;
}
if (n != 1) {
count *= 2;
}
return count;
}
 
BigInt OEISA073916(int n) {
if (isPrime(n) ) {
return BigInt(smallPrimes[n-1]) ^^ (n - 1);
}
int count = 0;
int result = 0;
for (int i = 1; count < n; i++) {
if (n % 2 == 1) {
// The solution for an odd (non-prime) term is always a square number
int root = cast(int) sqrt(cast(real) i);
if (root * root != i) {
continue;
}
}
if (getDivisorCount(i) == n) {
count++;
result = i;
}
}
return BigInt(result);
}
 
void main() {
foreach (n; 1 .. MAX + 1) {
writeln("A073916(", n, ") = ", OEISA073916(n));
}
}
Output:
A073916(1) = 1
A073916(2) = 3
A073916(3) = 25
A073916(4) = 14
A073916(5) = 14641
A073916(6) = 44
A073916(7) = 24137569
A073916(8) = 70
A073916(9) = 1089
A073916(10) = 405
A073916(11) = 819628286980801
A073916(12) = 160
A073916(13) = 22563490300366186081
A073916(14) = 2752
A073916(15) = 9801
A073916(16) = 462
A073916(17) = 21559177407076402401757871041
A073916(18) = 1044
A073916(19) = 740195513856780056217081017732809
A073916(20) = 1520
A073916(21) = 141376
A073916(22) = 84992
A073916(23) = 1658509762573818415340429240403156732495289
A073916(24) = 1170
A073916(25) = 52200625
A073916(26) = 421888
A073916(27) = 52900
A073916(28) = 9152
A073916(29) = 1116713952456127112240969687448211536647543601817400964721
A073916(30) = 6768
A073916(31) = 1300503809464370725741704158412711229899345159119325157292552449
A073916(32) = 3990
A073916(33) = 12166144
A073916(34) = 9764864
A073916(35) = 446265625
A073916(36) = 5472
A073916(37) = 11282036144040442334289838466416927162302790252609308623697164994458730076798801
A073916(38) = 43778048
A073916(39) = 90935296
A073916(40) = 10416
A073916(41) = 1300532588674810624476094551095787816112173600565095470117230812218524514342511947837104801
A073916(42) = 46400
A073916(43) = 635918448514386699807643535977466343285944704172890141356181792680152445568879925105775366910081
A073916(44) = 240640
A073916(45) = 327184

Factor[edit]

This makes use of most of the optimizations discussed in the Go example.

USING: combinators formatting fry kernel lists lists.lazy
lists.lazy.examples literals math math.functions math.primes
math.primes.factors math.ranges sequences ;
IN: rosetta-code.nth-n-div
 
CONSTANT: primes $[ 100 nprimes ]
 
: prime ( m -- n ) 1 - [ primes nth ] [ ^ ] bi ;
 
: (non-prime) ( m quot -- n )
'[
[ 1 - ] [ drop @ ] [ ] tri '[ divisors length _ = ]
lfilter swap [ cdr ] times car
] call ; inline
 
: non-prime ( m quot -- n )
{
{ [ over 2 = ] [ 2drop 3 ] }
{ [ over 10 = ] [ 2drop 405 ] }
[ (non-prime) ]
} cond ; inline
 
: fn ( m -- n )
{
{ [ dup even? ] [ [ evens ] non-prime ] }
{ [ dup prime? ] [ prime ] }
[ [ squares ] non-prime ]
} cond ;
 
: main ( -- ) 45 [1,b] [ dup fn "%2d : %d\n" printf ] each ;
 
MAIN: main
Output:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14641
 6 : 44
 7 : 24137569
 8 : 70
 9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
16 : 462
17 : 21559177407076402401757871041
18 : 1044
19 : 740195513856780056217081017732809
20 : 1520
21 : 141376
22 : 84992
23 : 1658509762573818415340429240403156732495289
24 : 1170
25 : 52200625
26 : 421888
27 : 52900
28 : 9152
29 : 1116713952456127112240969687448211536647543601817400964721
30 : 6768
31 : 1300503809464370725741704158412711229899345159119325157292552449
32 : 3990
33 : 12166144
34 : 9764864
35 : 446265625
36 : 5472
37 : 11282036144040442334289838466416927162302790252609308623697164994458730076798801
38 : 43778048
39 : 90935296
40 : 10416
41 : 1300532588674810624476094551095787816112173600565095470117230812218524514342511947837104801
42 : 46400
43 : 635918448514386699807643535977466343285944704172890141356181792680152445568879925105775366910081
44 : 240640
45 : 327184

Go[edit]

This makes use of the relationship: a[p] = prime[p]^(p-1) if p is prime, mentioned in the blurb for A073916 (and also on the talk page) to calculate the larger terms, some of which require big.Int in Go. It also makes use of another hint on the talk page that all odd terms are square numbers.

The remaining terms (up to the 33rd) are not particularly large and so are calculated by brute force.

package main
 
import (
"fmt"
"math"
"math/big"
)
 
var bi = new(big.Int)
 
func isPrime(n int) bool {
bi.SetUint64(uint64(n))
return bi.ProbablyPrime(0)
}
 
func generateSmallPrimes(n int) []int {
primes := make([]int, n)
primes[0] = 2
for i, count := 3, 1; count < n; i += 2 {
if isPrime(i) {
primes[count] = i
count++
}
}
return primes
}
 
func countDivisors(n int) int {
count := 1
for n%2 == 0 {
n >>= 1
count++
}
for d := 3; d*d <= n; d += 2 {
q, r := n/d, n%d
if r == 0 {
dc := 0
for r == 0 {
dc += count
n = q
q, r = n/d, n%d
}
count += dc
}
}
if n != 1 {
count *= 2
}
return count
}
 
func main() {
const max = 33
primes := generateSmallPrimes(max)
z := new(big.Int)
p := new(big.Int)
fmt.Println("The first", max, "terms in the sequence are:")
for i := 1; i <= max; i++ {
if isPrime(i) {
z.SetUint64(uint64(primes[i-1]))
p.SetUint64(uint64(i - 1))
z.Exp(z, p, nil)
fmt.Printf("%2d : %d\n", i, z)
} else {
count := 0
for j := 1; ; j++ {
if i%2 == 1 {
sq := int(math.Sqrt(float64(j)))
if sq*sq != j {
continue
}
}
if countDivisors(j) == i {
count++
if count == i {
fmt.Printf("%2d : %d\n", i, j)
break
}
}
}
}
}
}
Output:
The first 33 terms in the sequence are:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14641
 6 : 44
 7 : 24137569
 8 : 70
 9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
16 : 462
17 : 21559177407076402401757871041
18 : 1044
19 : 740195513856780056217081017732809
20 : 1520
21 : 141376
22 : 84992
23 : 1658509762573818415340429240403156732495289
24 : 1170
25 : 52200625
26 : 421888
27 : 52900
28 : 9152
29 : 1116713952456127112240969687448211536647543601817400964721
30 : 6768
31 : 1300503809464370725741704158412711229899345159119325157292552449
32 : 3990
33 : 12166144

The following much faster version (runs in less than 90 seconds on my 1.6GHz Celeron) uses three further optimizations:

1. Apart from the 2nd and 10th terms, all the even terms are themselves even.

2. A sieve is used to generate all prime divisors needed. This doesn't take up much time or memory but speeds up the counting of all divisors considerably.

3. While searching for the nth number with exactly n divisors, where feasible a record is kept of any numbers found to have exactly k divisors (k > n) so that the search for these numbers can start from a higher base.

package main
 
import (
"fmt"
"math"
"math/big"
)
 
type record struct{ num, count int }
 
var (
bi = new(big.Int)
primes = []int{2}
)
 
func isPrime(n int) bool {
bi.SetUint64(uint64(n))
return bi.ProbablyPrime(0)
}
 
func sieve(limit int) {
c := make([]bool, limit+1) // composite = true
// no need to process even numbers
p := 3
for {
p2 := p * p
if p2 > limit {
break
}
for i := p2; i <= limit; i += 2 * p {
c[i] = true
}
for {
p += 2
if !c[p] {
break
}
}
}
for i := 3; i <= limit; i += 2 {
if !c[i] {
primes = append(primes, i)
}
}
}
 
func countDivisors(n int) int {
count := 1
for i, p := 0, primes[0]; p*p <= n; i, p = i+1, primes[i+1] {
if n%p != 0 {
continue
}
n /= p
count2 := 1
for n%p == 0 {
n /= p
count2++
}
count *= (count2 + 1)
if n == 1 {
return count
}
}
if n != 1 {
count *= 2
}
return count
}
 
func isOdd(x int) bool {
return x%2 == 1
}
 
func main() {
sieve(22000)
const max = 45
records := [max + 1]record{}
z := new(big.Int)
p := new(big.Int)
fmt.Println("The first", max, "terms in the sequence are:")
for i := 1; i <= max; i++ {
if isPrime(i) {
z.SetUint64(uint64(primes[i-1]))
p.SetUint64(uint64(i - 1))
z.Exp(z, p, nil)
fmt.Printf("%2d : %d\n", i, z)
} else {
count := records[i].count
if count == i {
fmt.Printf("%2d : %d\n", i, records[i].num)
continue
}
odd := isOdd(i)
k := records[i].num
l := 1
if !odd && i != 2 && i != 10 {
l = 2
}
for j := k + l; ; j += l {
if odd {
sq := int(math.Sqrt(float64(j)))
if sq*sq != j {
continue
}
}
cd := countDivisors(j)
if cd == i {
count++
if count == i {
fmt.Printf("%2d : %d\n", i, j)
break
}
} else if cd > i && cd <= max && records[cd].count < cd &&
j > records[cd].num && (l == 1 || (l == 2 && !isOdd(cd))) {
records[cd].num = j
records[cd].count++
}
}
}
}
}
Output:
The first 45 terms in the sequence are:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14641
 6 : 44
 7 : 24137569
 8 : 70
 9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
16 : 462
17 : 21559177407076402401757871041
18 : 1044
19 : 740195513856780056217081017732809
20 : 1520
21 : 141376
22 : 84992
23 : 1658509762573818415340429240403156732495289
24 : 1170
25 : 52200625
26 : 421888
27 : 52900
28 : 9152
29 : 1116713952456127112240969687448211536647543601817400964721
30 : 6768
31 : 1300503809464370725741704158412711229899345159119325157292552449
32 : 3990
33 : 12166144
34 : 9764864
35 : 446265625
36 : 5472
37 : 11282036144040442334289838466416927162302790252609308623697164994458730076798801
38 : 43778048
39 : 90935296
40 : 10416
41 : 1300532588674810624476094551095787816112173600565095470117230812218524514342511947837104801
42 : 46400
43 : 635918448514386699807643535977466343285944704172890141356181792680152445568879925105775366910081
44 : 240640
45 : 327184

Haskell[edit]

import           Control.Monad                         (guard)
import Math.NumberTheory.ArithmeticFunctions (divisorCount)
import Math.NumberTheory.Primes (Prime, unPrime)
import Math.NumberTheory.Primes.Testing (isPrime)
 
calc :: Integer -> [(Integer, Integer)]
calc n = do
x <- [1..]
guard (even n || odd n && f x == x)
[(x, divisorCount x)]
where f n = floor (sqrt $ realToFrac n) ^ 2
 
havingNthDivisors :: Integer -> [(Integer, Integer)]
havingNthDivisors n = filter ((==n) . snd) $ calc n
 
nths :: [(Integer, Integer)]
nths = do
n <- [1..35] :: [Integer]
if isPrime n then
pure (n, nthPrime (fromIntegral n) ^ pred n)
else
pure (n, f n)
where
f n = fst (havingNthDivisors n !! pred (fromIntegral n))
nthPrime n = unPrime (toEnum n :: Prime Integer)
 
main :: IO ()
main = mapM_ print nths
Output:
(1,1)
(2,3)
(3,25)
(4,14)
(5,14641)
(6,44)
(7,24137569)
(8,70)
(9,1089)
(10,405)
(11,819628286980801)
(12,160)
(13,22563490300366186081)
(14,2752)
(15,9801)
(16,462)
(17,21559177407076402401757871041)
(18,1044)
(19,740195513856780056217081017732809)
(20,1520)
(21,141376)
(22,84992)
(23,1658509762573818415340429240403156732495289)
(24,1170)
(25,52200625)
(26,421888)
(27,52900)
(28,9152)
(29,1116713952456127112240969687448211536647543601817400964721)
(30,6768)
(31,1300503809464370725741704158412711229899345159119325157292552449)
(32,3990)
(33,12166144)
(34,9764864)
(35,446265625)

Java[edit]

Replace translation with Java native implementation.

 
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;
 
public class SequenceNthNumberWithExactlyNDivisors {
 
public static void main(String[] args) {
int max = 45;
smallPrimes(max);
for ( int n = 1; n <= max ; n++ ) {
System.out.printf("A073916(%d) = %s%n", n, OEISA073916(n));
}
}
 
private static List<Integer> smallPrimes = new ArrayList<>();
 
private static void smallPrimes(int numPrimes) {
smallPrimes.add(2);
for ( int n = 3, count = 0 ; count < numPrimes ; n += 2 ) {
if ( isPrime(n) ) {
smallPrimes.add(n);
count++;
}
}
}
 
private static final boolean isPrime(long test) {
if ( test == 2 ) {
return true;
}
if ( test % 2 == 0 ) {
return false;
}
for ( long d = 3 ; d*d <= test ; d += 2 ) {
if ( test % d == 0 ) {
return false;
}
}
return true;
}
 
private static int getDivisorCount(long n) {
int count = 1;
while ( n % 2 == 0 ) {
n /= 2;
count += 1;
}
for ( long d = 3 ; d*d <= n ; d += 2 ) {
long q = n / d;
long r = n % d;
int dc = 0;
while ( r == 0 ) {
dc += count;
n = q;
q = n / d;
r = n % d;
}
count += dc;
}
if ( n != 1 ) {
count *= 2;
}
return count;
}
 
private static BigInteger OEISA073916(int n) {
if ( isPrime(n) ) {
return BigInteger.valueOf(smallPrimes.get(n-1)).pow(n - 1);
}
int count = 0;
int result = 0;
for ( int i = 1 ; count < n ; i++ ) {
if ( n % 2 == 1 ) {
// The solution for an odd (non-prime) term is always a square number
int sqrt = (int) Math.sqrt(i);
if ( sqrt*sqrt != i ) {
continue;
}
}
if ( getDivisorCount(i) == n ) {
count++;
result = i;
}
}
return BigInteger.valueOf(result);
}
 
}
 
Output:
A073916(1) = 1
A073916(2) = 3
A073916(3) = 25
A073916(4) = 14
A073916(5) = 14641
A073916(6) = 44
A073916(7) = 24137569
A073916(8) = 70
A073916(9) = 1089
A073916(10) = 405
A073916(11) = 819628286980801
A073916(12) = 160
A073916(13) = 22563490300366186081
A073916(14) = 2752
A073916(15) = 9801
A073916(16) = 462
A073916(17) = 21559177407076402401757871041
A073916(18) = 1044
A073916(19) = 740195513856780056217081017732809
A073916(20) = 1520
A073916(21) = 141376
A073916(22) = 84992
A073916(23) = 1658509762573818415340429240403156732495289
A073916(24) = 1170
A073916(25) = 52200625
A073916(26) = 421888
A073916(27) = 52900
A073916(28) = 9152
A073916(29) = 1116713952456127112240969687448211536647543601817400964721
A073916(30) = 6768
A073916(31) = 1300503809464370725741704158412711229899345159119325157292552449
A073916(32) = 3990
A073916(33) = 12166144
A073916(34) = 9764864
A073916(35) = 446265625
A073916(36) = 5472
A073916(37) = 11282036144040442334289838466416927162302790252609308623697164994458730076798801
A073916(38) = 43778048
A073916(39) = 90935296
A073916(40) = 10416
A073916(41) = 1300532588674810624476094551095787816112173600565095470117230812218524514342511947837104801
A073916(42) = 46400
A073916(43) = 635918448514386699807643535977466343285944704172890141356181792680152445568879925105775366910081
A073916(44) = 240640
A073916(45) = 327184

jq[edit]

Adapted from Wren
Works with gojq, the Go implementation of jq

See Erdős-primes#jq for a suitable implementation of `is_prime`.

The precision of the integer arithmetic of the C implementation of jq is only precise enough for computing the n-th value up to and including [16,462]. Accordingly gojq was used to produce the output shown below.

Preliminaries

def count(stream): reduce stream as $i (0; .+1);
 
# To maintain precision:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);
 
def primes: 2, (range(3; infinite; 2) | select(is_prime));
 
# divisors as an unsorted stream
def divisors:
if . == 1 then 1
else . as $n
| label $out
| range(1; $n) as $i
| ($i * $i) as $i2
| if $i2 > $n then break $out
else if $i2 == $n
then $i
elif ($n % $i) == 0
then $i, ($n/$i)
else empty
end
end
end;
 

The Task

# Emit [n, nth_with_n_divisors] for n in range(1; .+1)
def nth_with_n_divisors:
| [limit( .; primes)] as $primes
| range( 1; 1 + .) as $i
| if $i | is_prime
then [$i, ($primes[$i-1]|power($i-1))]
else {count: 0, j: 1}
| until(.count == $i ;
.cont = false
| if ($i % 2) == 1 then (.j|sqrt|floor) as $sq
| if ($sq * $sq) != .j then .j += 1 | .cont = true else . end
else .
end
| if .cont == false
then if (.j | count(divisors)) == $i
then .count += 1
else .
end
| if .count != $i then .j += 1 else . end
else .
end )
 
| [ $i, .j]
end;
 
"The first 33 terms in the sequence are:",
(33 | nth_with_n_divisors)
Output:
The first 33 terms in the sequence are:
[1,1]
[2,3]
[3,25]
[4,14]
[5,14641]
[6,44]
[7,24137569]
[8,70]
[9,1089]
[10,405]
[11,819628286980801]
[12,160]
[13,22563490300366186081]
[14,2752]
[15,9801]
[16,462]
[17,21559177407076402401757871041]
[18,1044]
[19,740195513856780056217081017732809]
[20,1520]
[21,141376]
[22,84992]
[23,1658509762573818415340429240403156732495289]
[24,1170]
[25,52200625]
[26,421888]
[27,52900]
[28,9152]
[29,1116713952456127112240969687448211536647543601817400964721]
[30,6768]
[31,1300503809464370725741704158412711229899345159119325157292552449]
[32,3990]
[33,12166144]

Julia[edit]

using Primes
 
function countdivisors(n)
f = [one(n)]
for (p, e) in factor(n)
f = reduce(vcat, [f * p ^ j for j in 1:e], init = f)
end
length(f)
end
 
function nthwithndivisors(N)
parray = findall(primesmask(100 * N))
for i = 1:N
if isprime(i)
println("$i : ", BigInt(parray[i])^(i-1))
else
k = 0
for j in 1:100000000000
if (iseven(i) || Int(floor(sqrt(j)))^2 == j) &&
i == countdivisors(j) && (k += 1) == i
println("$i : $j")
break
end
end
end
end
end
 
nthwithndivisors(35)
 
Output:
1 : 1
2 : 3
3 : 25
4 : 14
5 : 14641
6 : 44
7 : 24137569
8 : 70
9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
16 : 462
17 : 21559177407076402401757871041
18 : 1044
19 : 740195513856780056217081017732809
20 : 1520
21 : 141376
22 : 84992
23 : 1658509762573818415340429240403156732495289
24 : 1170
25 : 52200625
26 : 421888
27 : 52900
28 : 9152
29 : 1116713952456127112240969687448211536647543601817400964721
30 : 6768
31 : 1300503809464370725741704158412711229899345159119325157292552449
32 : 3990
33 : 12166144
34 : 9764864
35 : 446265625

Kotlin[edit]

Translation of: Go
// Version 1.3.21
 
import java.math.BigInteger
import kotlin.math.sqrt
 
const val MAX = 33
 
fun isPrime(n: Int) = BigInteger.valueOf(n.toLong()).isProbablePrime(10)
 
fun generateSmallPrimes(n: Int): List<Int> {
val primes = mutableListOf<Int>()
primes.add(2)
var i = 3
while (primes.size < n) {
if (isPrime(i)) {
primes.add(i)
}
i += 2
}
return primes
}
 
fun countDivisors(n: Int): Int {
var nn = n
var count = 1
while (nn % 2 == 0) {
nn = nn shr 1
count++
}
var d = 3
while (d * d <= nn) {
var q = nn / d
var r = nn % d
if (r == 0) {
var dc = 0
while (r == 0) {
dc += count
nn = q
q = nn / d
r = nn % d
}
count += dc
}
d += 2
}
if (nn != 1) count *= 2
return count
}
 
fun main() {
var primes = generateSmallPrimes(MAX)
println("The first $MAX terms in the sequence are:")
for (i in 1..MAX) {
if (isPrime(i)) {
var z = BigInteger.valueOf(primes[i - 1].toLong())
z = z.pow(i - 1)
System.out.printf("%2d : %d\n", i, z)
} else {
var count = 0
var j = 1
while (true) {
if (i % 2 == 1) {
val sq = sqrt(j.toDouble()).toInt()
if (sq * sq != j) {
j++
continue
}
}
if (countDivisors(j) == i) {
if (++count == i) {
System.out.printf("%2d : %d\n", i, j)
break
}
}
j++
}
}
}
}
Output:
The first 33 terms in the sequence are:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14641
 6 : 44
 7 : 24137569
 8 : 70
 9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
16 : 462
17 : 21559177407076402401757871041
18 : 1044
19 : 740195513856780056217081017732809
20 : 1520
21 : 141376
22 : 84992
23 : 1658509762573818415340429240403156732495289
24 : 1170
25 : 52200625
26 : 421888
27 : 52900
28 : 9152
29 : 1116713952456127112240969687448211536647543601817400964721
30 : 6768
31 : 1300503809464370725741704158412711229899345159119325157292552449
32 : 3990
33 : 12166144

Mathematica / Wolfram Language[edit]

d = Table[
Length[Divisors[n]], {n,
200000}]; t = {}; n = 0; ok = True; While[ok, n++;
If[PrimeQ[n], AppendTo[t, Prime[n]^(n - 1)],
c = Flatten[Position[d, n, 1, n]];
If[Length[c] >= n, AppendTo[t, c[[n]]], ok = False]]];
t
Output:
{1,3,25,14,14641,44,24137569,70,1089,405,819628286980801,160,22563490300366186081,2752,9801,462,21559177407076402401757871041,1044,740195513856780056217081017732809,1520,141376,84992,1658509762573818415340429240403156732495289,1170}

Nim[edit]

Translation of: Go
Library: bignum

This is a translation of the fast Go version. It runs in about 23s on our laptop.

import math, strformat
import bignum
 
type Record = tuple[num, count: Natural]
 
template isOdd(n: Natural): bool =
(n and 1) != 0
 
func isPrime(n: int): bool =
let bi = newInt(n)
result = bi.probablyPrime(25) != 0
 
proc findPrimes(limit: Natural): seq[int] {.compileTime.} =
result = @[2]
var isComposite = newSeq[bool](limit + 1)
var p = 3
while true:
let p2 = p * p
if p2 > limit: break
for i in countup(p2, limit, 2 * p):
isComposite[i] = true
while true:
inc p, 2
if not isComposite[p]: break
for n in countup(3, limit, 2):
if not isComposite[n]:
result.add n
 
const Primes = findPrimes(22_000)
 
proc countDivisors(n: Natural): int =
result = 1
var n = n
for i, p in Primes:
if p * p > n: break
if n mod p != 0: continue
n = n div p
var count = 1
while n mod p == 0:
n = n div p
inc count
result *= count + 1
if n == 1: return
if n != 1: result *= 2
 
const Max = 45
var records: array[0..Max, Record]
echo &"The first {Max} terms in the sequence are:"
 
for n in 1..Max:
 
if n.isPrime:
var z = newInt(Primes[n - 1])
z = pow(z, culong(n - 1))
echo &"{n:2}: {z}"
 
else:
var count = records[n].count
if count == n:
echo &"{n:2}: {records[n].num}"
continue
let odd = n.isOdd
let d = if odd or n == 2 or n == 10: 1 else: 2
var k = records[n].num
while true:
inc k, d
if odd:
let sq = sqrt(k.toFloat).int
if sq * sq != k: continue
let cd = k.countDivisors()
if cd == n:
inc count
if count == n:
echo &"{n:2}: {k}"
break
elif cd in (n + 1)..Max and records[cd].count < cd and
k > records[cd].num and (d == 1 or d == 2 and not cd.isOdd):
records[cd].num = k
inc records[cd].count
Output:
The first 45 terms in the sequence are:
 1: 1
 2: 3
 3: 25
 4: 14
 5: 14641
 6: 44
 7: 24137569
 8: 70
 9: 1089
10: 405
11: 819628286980801
12: 160
13: 22563490300366186081
14: 2752
15: 9801
16: 462
17: 21559177407076402401757871041
18: 1044
19: 740195513856780056217081017732809
20: 1520
21: 141376
22: 84992
23: 1658509762573818415340429240403156732495289
24: 1170
25: 52200625
26: 421888
27: 52900
28: 9152
29: 1116713952456127112240969687448211536647543601817400964721
30: 6768
31: 1300503809464370725741704158412711229899345159119325157292552449
32: 3990
33: 12166144
34: 9764864
35: 446265625
36: 5472
37: 11282036144040442334289838466416927162302790252609308623697164994458730076798801
38: 43778048
39: 90935296
40: 10416
41: 1300532588674810624476094551095787816112173600565095470117230812218524514342511947837104801
42: 46400
43: 635918448514386699807643535977466343285944704172890141356181792680152445568879925105775366910081
44: 240640
45: 327184

Perl[edit]

Library: ntheory
Translation of: Raku
use strict;
use warnings;
use bigint;
use ntheory <nth_prime is_prime divisors>;
 
my $limit = 20;
 
print "First $limit terms of OEIS:A073916\n";
 
for my $n (1..$limit) {
if ($n > 4 and is_prime($n)) {
print nth_prime($n)**($n-1) . ' ';
} else {
my $i = my $x = 0;
while (1) {
my $nn = $n%2 ? ++$x**2 : ++$x;
next unless $n == divisors($nn) and ++$i == $n;
print "$nn " and last;
}
}
}
Output:
First 20 terms of OEIS:A073916
1 3 25 14 14641 44 24137569 70 1089 405 819628286980801 160 22563490300366186081 2752 9801 462 21559177407076402401757871041 1044 740195513856780056217081017732809 1520

Phix[edit]

Library: Phix/mpfr

simple[edit]

Certainly not the fastest way to do it, hence the relatively small limit of 24, which takes less than 0.4s,
whereas a limit of 25 would need to invoke factors() 52 million times which would no doubt take a fair while.

with javascript_semantics 
constant LIMIT = 24
include mpfr.e
mpz z = mpz_init()
 
sequence fn = repeat(0,LIMIT) fn[1] = 1
integer k = 1
printf(1,"The first %d terms in the sequence are:\n",LIMIT)
for i=1 to LIMIT do
    if is_prime(i) then
        mpz_ui_pow_ui(z,get_prime(i),i-1)
        printf(1,"%2d : %s\n",{i,mpz_get_str(z)})
    else
        while fn[i]<i do
            k += 1
            integer l = length(factors(k,1))
            if l<=LIMIT and fn[l]<l then
                fn[l] = iff(fn[l]+1<l?fn[l]+1:k)
            end if
        end while
        printf(1,"%2d : %d\n",{i,fn[i]})
    end if
end for
Output:
The first 24 terms in the sequence are:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14641
 6 : 44
 7 : 24137569
 8 : 70
 9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
16 : 462
17 : 21559177407076402401757871041
18 : 1044
19 : 740195513856780056217081017732809
20 : 1520
21 : 141376
22 : 84992
23 : 1658509762573818415340429240403156732495289
24 : 1170

cheating slightly[edit]

No real patterns that I could see here, but you can still identify and single out the troublemakers (of which there are about 30).

with javascript_semantics 
include mpfr.e
atom t0 = time()
constant LIMIT = 100
include mpfr.e
include primes.e
mpz z = mpz_init(),
    p = mpz_init() 
string mz
sequence fn = repeat(0,LIMIT), dx;  fn[1] = 1
integer k = 1, idx, p1, p2
printf(1,"The first %d terms in the sequence are:\n",LIMIT)
for i=1 to LIMIT do
    if is_prime(i) or i=1 then
        mpz_ui_pow_ui(z,get_prime(i),i-1)
        mz = mpz_get_str(z)
    else
        sequence f = prime_factors(i,1)
        if length(f)=2 and f[1]=2 and f[2]>7 then
            mz = sprintf("%d",power(2,f[2]-1)*get_prime(i+1))
        elsif length(f)=2 and f[1]>2 then
            if f[1]=f[2] then
                mz = sprintf("%d",power(f[1]*get_prime(f[1]+2),f[1]-1))
            else -- deal with some tardy ones...
                dx = {15,21,33,35,39,51,55,57,65,69,77,85,87,91,93,95}; idx = find(i,dx)
                p1 = { 3, 2, 2, 5, 2, 2, 2, 2, 2, 2, 7, 2, 2, 7, 2, 2}[idx]
                p2 = { 5,15,29, 6,35,49,34,56,45,69, 7,65,88, 7,94,77}[idx]
                mpz_ui_pow_ui(z,p1,f[2]-1)
                mpz_ui_pow_ui(p,get_prime(p2),f[1]-1)
                mpz_mul(z,z,p)
                mz = mpz_get_str(z)
            end if
        elsif (length(f)=3 and i>50) or (length(f)=4 and (f[1]=3 or f[4]>7)) then 
            if i=99 then    -- (oops, messed that one up!)
                mz = sprintf("%d",4*power(3,10)*31*31)
            elsif i=63 then -- (and another!)
                mz = sprintf("%d",power(2,8)*power(5,6))
            else
                dx = {52,66,68,70,75,76,78,92,98,81,88}; idx = find(i,dx)
                p1 = { 7, 3, 1, 5, 3, 5, 5,13, 3,35,35}[idx]
                p2 = { 1, 2, 1, 4, 4, 1, 2, 1, 1, 2, 1}[idx]
                mpz_ui_pow_ui(z,2,f[$]-1)
                mpz_ui_pow_ui(p,p1,p2)
                mpz_mul(z,z,p)
                p1 = {13,37, 4, 9,34,22,19,12, 4,11,13}[idx]
                p2 = { 1, 1, 3, 1, 2, 1, 1, 1, 6, 2, 1}[idx]
                mpz_ui_pow_ui(p,get_prime(p1),p2)
                mpz_mul(z,z,p)
                mz = mpz_get_str(z)
            end if
        else
            while fn[i]<i do
                k += 1
                integer l = length(factors(k,1))
                if l<=LIMIT and fn[l]<l then
                    fn[l] = iff(fn[l]+1<l?fn[l]+1:k)
                end if
            end while
            mz = sprintf("%d",fn[i])
        end if
    end if
    printf(1,"%3d : %s\n",{i,mz})
end for
printf(1,"completed in %s\n",{elapsed(time()-t0)})
Output:
The first 100 terms in the sequence are:
  1 : 1
  2 : 3
  3 : 25
  4 : 14
  5 : 14641
  6 : 44
  7 : 24137569
  8 : 70
  9 : 1089
 10 : 405
 11 : 819628286980801
 12 : 160
 13 : 22563490300366186081
 14 : 2752
 15 : 9801
 16 : 462
 17 : 21559177407076402401757871041
 18 : 1044
 19 : 740195513856780056217081017732809
 20 : 1520
 21 : 141376
 22 : 84992
 23 : 1658509762573818415340429240403156732495289
 24 : 1170
 25 : 52200625
 26 : 421888
 27 : 52900
 28 : 9152
 29 : 1116713952456127112240969687448211536647543601817400964721
 30 : 6768
 31 : 1300503809464370725741704158412711229899345159119325157292552449
 32 : 3990
 33 : 12166144
 34 : 9764864
 35 : 446265625
 36 : 5472
 37 : 11282036144040442334289838466416927162302790252609308623697164994458730076798801
 38 : 43778048
 39 : 90935296
 40 : 10416
 41 : 1300532588674810624476094551095787816112173600565095470117230812218524514342511947837104801
 42 : 46400
 43 : 635918448514386699807643535977466343285944704172890141356181792680152445568879925105775366910081
 44 : 240640
 45 : 327184
 46 : 884998144
 47 : 82602452843197830915655434062758747152610200533183747995128511868250464749389571755574391210629602061883161
 48 : 10296
 49 : 17416274304961
 50 : 231984
 51 : 3377004544
 52 : 1175552
 53 : 7326325566540660915295202005885275873916026034616342139474905237555535331121749053330837020397976615915057535109963186790081
 54 : 62208
 55 : 382260265984
 56 : 63168
 57 : 18132238336
 58 : 74356621312
 59 : 4611334279555550707926152839105934955536765902552873727962394200823974159354935875908492026570361080937000929065119751494662472171586496615769
 60 : 37200
 61 : 1279929743416851311019131209907830943453757487243270654630811620734985849511676634764875391422075025095805774223361200187655617244608064273703030801
 62 : 329638739968
 63 : 4000000
 64 : 41160
 65 : 6169143218176
 66 : 1446912
 67 : 20353897784481135224502113429729640062994484338530413467091588021107086251737634020247647652000753728181181145357697865506347474542010115076391004870941216126804332281
 68 : 22478848
 69 : 505031950336
 70 : 920000
 71 : 22091712217028661091647719716134154062183987922906664635563029317259865249987461330814689139636373404600637581380931231750650949001643115899851798743405544731506806491024751606849
 72 : 48300
 73 : 45285235038445046669368642612544904396805516154393281169675637706411327508046898517381759728413013085702957690245765106506995874808813788844198933536768701568785385215106907990288684161
 74 : 26044681682944
 75 : 25040016
 76 : 103546880
 77 : 6818265813529681
 78 : 6860800
 79 : 110984176612396876252402058909207317796166059426692518840795949938301678339569859458072604697803922487329059012193474923358078243829751108364014428972188856355641430510895584045477184112155202949344511201
 80 : 96720
 81 : 4708900
 82 : 473889511571456
 83 : 1064476683917919713953093000677954858036756167846865592483240200233630032347646244510522542053167377047784795269272961130616738371982635464615430562192693194769301221853619917764723198332349478419665523610384617408161
 84 : 225216
 85 : 629009610244096
 86 : 1974722883485696
 87 : 56062476550144
 88 : 1469440
 89 : 2544962774801294304714624882135254894108219227449639770372304502957346499018390075803907657903246999131414158076182409047363202723848127272231619125736007088495905384436604400674375401897829996007586872027878808309385140119563002941281
 90 : 352512
 91 : 334095024862954369
 92 : 2017460224
 93 : 258858752671744
 94 : 35114003344654336
 95 : 6002585119227904
 96 : 112860
 97 : 69969231567692157576407845029145070949540195647704307603423555494283752374775631665902846216473259715737953596002226233187827382886325202177640164868195792546734599315840795700630834939445407388277880586442087150607690134279001258366485550281200590593848327041
 98 : 22588608
 99 : 226984356
100 : 870000
completed in 4.4s

Python[edit]

This implementation exploits the fact that terms corresponding to a prime value for n are always the nth prime to the (n-1)th power.

 
def divisors(n):
divs = [1]
for ii in range(2, int(n ** 0.5) + 3):
if n % ii == 0:
divs.append(ii)
divs.append(int(n / ii))
divs.append(n)
return list(set(divs))
 
 
def is_prime(n):
return len(divisors(n)) == 2
 
 
def primes():
ii = 1
while True:
ii += 1
if is_prime(ii):
yield ii
 
 
def prime(n):
generator = primes()
for ii in range(n - 1):
generator.__next__()
return generator.__next__()
 
 
def n_divisors(n):
ii = 0
while True:
ii += 1
if len(divisors(ii)) == n:
yield ii
 
 
def sequence(max_n=None):
if max_n is not None:
for ii in range(1, max_n + 1):
if is_prime(ii):
yield prime(ii) ** (ii - 1)
else:
generator = n_divisors(ii)
for jj, out in zip(range(ii - 1), generator):
pass
yield generator.__next__()
else:
ii = 1
while True:
ii += 1
if is_prime(ii):
yield prime(ii) ** (ii - 1)
else:
generator = n_divisors(ii)
for jj, out in zip(range(ii - 1), generator):
pass
yield generator.__next__()
 
 
if __name__ == '__main__':
for item in sequence(15):
print(item)
 

Output:

 
1
3
25
14
14641
44
24137569
70
1089
405
819628286980801
160
22563490300366186081
2752
9801
 

Raku[edit]

(formerly Perl 6)

Works with: Rakudo version 2019.03

Try it online!

sub div-count (\x) {
return 2 if x.is-prime;
+flat (1 .. x.sqrt.floor).map: -> \d {
unless x % d { my \y = x div d; y == d ?? y !! (y, d) }
}
}
 
my $limit = 20;
 
my @primes = grep { .is-prime }, 1..*;
@primes[$limit]; # prime the array. SCNR
 
put "First $limit terms of OEIS:A073916";
put (1..$limit).hyper(:2batch).map: -> $n {
($n > 4 and $n.is-prime) ??
exp($n - 1, @primes[$n - 1]) !!
do {
my $i = 0;
my $iterator = $n %% 2 ?? (1..*) !! (1..*).map: *²;
$iterator.first: {
next unless $n == .&div-count;
next unless ++$i == $n;
$_
}
}
};
First 20 terms of OEIS:A073916
1 3 25 14 14641 44 24137569 70 1089 405 819628286980801 160 22563490300366186081 2752 9801 462 21559177407076402401757871041 1044 740195513856780056217081017732809 1520

REXX[edit]

Programming note:   this REXX version has minor optimization, and all terms of the sequence are determined (found) in order.

little optimization[edit]

/*REXX program  finds and displays  the    Nth  number   with exactly   N   divisors.   */
parse arg N . /*obtain optional argument from the CL.*/
if N=='' | N=="," then N= 15 /*Not specified? Then use the default.*/
if N>=50 then numeric digits 10 /*use more decimal digits for large N. */
w= 50 /*W: width of the 2nd column of output*/
say '─divisors─' center("the Nth number with exactly N divisors", w, '─') /*title.*/
@.1= 2; Ps= 1 /*1st prime; number of primes (so far)*/
do p=3 until Ps==N /* [↓] gen N primes, store in @ array.*/
if \isPrime(p) then iterate; Ps= Ps + 1; @.Ps= p
end /*gp*/
!.= /*the  ! array is used for memoization*/
do i=1 for N; odd= i//2 /*step through a number of divisors. */
if odd then if isPrime(i) then do; _= pPow(); w= max(w, length(_) )
call tell commas(_); iterate
end
#= 0; even= \odd /*the number of occurrences for #div. */
do j=1; jj= j /*now, search for a number that ≡ #divs*/
if odd then jj= j*j /*Odd and non-prime? Calculate square.*/
if !.jj==. then iterate /*has this number already been found? */
d= #divs(jj) /*get # divisors; Is not equal? Skip.*/
if even then if d<i then do;  !.j=.; iterate; end /*Too low? Flag it.*/
if d\==i then iterate /*Is not equal? Then skip this number.*/
#= # + 1 /*bump number of occurrences for #div. */
if #\==i then iterate /*Not correct occurrence? Keep looking.*/
call tell commas(jj) /*display Nth number with #divs*/
leave /*found a number, so now get the next I*/
end /*j*/
end /*i*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _; do j=length(_)-3 to 1 by -3; _=insert(',', _, j); end; return _
pPow: numeric digits 1000; return @.i**(i-1) /*temporarily increase decimal digits. */
tell: parse arg _; say center(i,10) right(_,max(w,length(_))); if i//5==0 then say; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
#divs: procedure; parse arg x 1 y /*X and Y: both set from 1st argument.*/
if x<7 then do /*handle special cases for numbers < 7.*/
if x<3 then return x /* " " " " one and two.*/
if x<5 then return x - 1 /* " " " " three & four*/
if x==5 then return 2 /* " " " " five. */
if x==6 then return 4 /* " " " " six. */
end
odd= x // 2 /*check if X is odd or not. */
if odd then do; #= 1; end /*Odd? Assume Pdivisors count of 1.*/
else do; #= 3; y= x%2; end /*Even? " " " " 3.*/
/* [↑] start with known num of Pdivs.*/
do k=3 by 1+odd while k<y /*when doing odd numbers, skip evens. */
if x//k==0 then do /*if no remainder, then found a divisor*/
#=#+2; y=x%k /*bump # Pdivs, calculate limit Y. */
if k>=y then do; #= #-1; leave; end /*limit?*/
end /* ___ */
else if k*k>x then leave /*only divide up to √ x */
end /*k*/ /* [↑] this form of DO loop is faster.*/
return #+1 /*bump "proper divisors" to "divisors".*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
isPrime: procedure; parse arg #; if wordpos(#, '2 3 5 7 11 13')\==0 then return 1
if #<2 then return 0; if #//2==0 | #//3==0 | #//5==0 | #//7==0 then return 0
if # // 2==0 | # // 3 ==0 then return 0
do j=11 by 6 until j*j>#; if # // j==0 | # // (J+2)==0 then return 0
end /*j*/ /* ___ */
return 1 /*Exceeded √ #  ? Then # is prime. */
output   when using the input:     45

(Shown at   3/4   size.)

─divisors─ ───────────────────────────────────────────the Nth number with exactly N divisors──────────────────────────────────────────────
    1                                                                                                                                    1
    2                                                                                                                                    3
    3                                                                                                                                   25
    4                                                                                                                                   14
    5                                                                                                                               14,641

    6                                                                                                                                   44
    7                                                                                                                           24,137,569
    8                                                                                                                                   70
    9                                                                                                                                1,089
    10                                                                                                                                 405

    11                                                                                                                 819,628,286,980,801
    12                                                                                                                                 160
    13                                                                                                          22,563,490,300,366,186,081
    14                                                                                                                               2,752
    15                                                                                                                               9,801

    16                                                                                                                                 462
    17                                                                                              21,559,177,407,076,402,401,757,871,041
    18                                                                                                                               1,044
    19                                                                                         740,195,513,856,780,056,217,081,017,732,809
    20                                                                                                                               1,520

    21                                                                                                                             141,376
    22                                                                                                                              84,992
    23                                                                           1,658,509,762,573,818,415,340,429,240,403,156,732,495,289
    24                                                                                                                               1,170
    25                                                                                                                          52,200,625

    26                                                                                                                             421,888
    27                                                                                                                              52,900
    28                                                                                                                               9,152
    29                                                       1,116,713,952,456,127,112,240,969,687,448,211,536,647,543,601,817,400,964,721
    30                                                                                                                               6,768

    31                                               1,300,503,809,464,370,725,741,704,158,412,711,229,899,345,159,119,325,157,292,552,449
    32                                                                                                                               3,990
    33                                                                                                                          12,166,144
    34                                                                                                                           9,764,864
    35                                                                                                                         446,265,625

    36                                                                                                                               5,472
    37                          11,282,036,144,040,442,334,289,838,466,416,927,162,302,790,252,609,308,623,697,164,994,458,730,076,798,801
    38                                                                                                                          43,778,048
    39                                                                                                                          90,935,296
    40                                                                                                                              10,416

    41           1,300,532,588,674,810,624,476,094,551,095,787,816,112,173,600,565,095,470,117,230,812,218,524,514,342,511,947,837,104,801
    42                                                                                                                              46,400
    43     635,918,448,514,386,699,807,643,535,977,466,343,285,944,704,172,890,141,356,181,792,680,152,445,568,879,925,105,775,366,910,081
    44                                                                                                                             240,640
    45                                                                                                                             327,184

more optimization[edit]

Programming note:   this REXX version has major optimization, and the logic flow is:

  •   build a table of prime numbers (this also helps winnow the numbers being tested).
  •   the generation of the sequence is broken into three parts:
  •   odd prime numbers.
  •   odd non-prime numbers.
  •   even numbers.

This REXX version (unlike the 1st version),   only goes through the numbers once, instead of looking for numbers that have specific number of divisors.

/*REXX program  finds and displays  the    Nth  number   with exactly   N   divisors.   */
parse arg N . /*obtain optional argument from the CL.*/
if N=='' | N=="," then N= 15 /*Not specified? Then use the default.*/
if N>=50 then numeric digits 10 /*use more decimal digits for large N. */
@.1= 2; Ps= 1;  !.= 0;  !.1= 2 /*1st prime; number of primes (so far)*/
do p=3 until Ps==N**3 /* [↓] gen N primes, store in @ array.*/
if \isPrime(p) then iterate; Ps= Ps + 1; if Ps<=N then @.Ps= p;  !.p= 1
end /*p*/
 
zfin.= 0; zcnt. = 0; znum.1= 1; znum.2= 3 /*completed; index; count of items.*/
w= 50 /*──────────handle odd primes──────────*/
do j=3 by 2 to N; if \!.j then iterate /*Not prime? Then skip this odd number*/
zfin.j= 1; zcnt.j= j; znum.j= pPow(); /*compute # divisors for this odd prime*/
w= max(w, length( commas( znum.j) ) ) /*the last prime will be the biggest #.*/
end /*j*/ /*process a small number of primes ≤ N.*/
dd.=; mx= 200000 /*──────────handle odd non─primes──────*/
do j=3 by 2 to N; if !.j then iterate /*Is a prime? Then skip this odd prime*/
do sq=6; _= sq*sq /*step through squares starting at 36.*/
if dd._\=='' then d= dd._ /*maybe use a pre─computed # divisors. */
else d= #divs(_) /*Not defined? Then calculate # divs. */
if _<=mx then dd._= d /*use memoization for the evens loop.*/
if d\==j then iterate /*if not the right D, then skip this sq*/
zcnt.d= zcnt.d+1; if zcnt.d==d then zfin.d= 1; znum.d= _
if zfin.d then iterate j /*if all were found, then do next odd#*/
end /*sq*/
end /*j*/
/*──────────handle even numbers.───────*/
do j=4 by 2; if dd.j\=='' then d= dd.j /*maybe use a pre─computed # divisors. */
else d= #divs(j) /*Not defined? Then calculate # divs. */
if d>N then iterate /*Divisors greater than N? Then skip. */
if zfin.d then iterate /*Already populated? " " */
else do; zcnt.d= zcnt.d+1; if zcnt.d==d then zfin.d= 1; znum.d= j
if done() then leave /*j*/ /*Are the even #'s all done? */
end
end /*j*/
 
say '─divisors─' center("the Nth number with exactly N divisors", w, '─') /*title.*/
do s=1 for N; call tell s,commas(znum.s) /*display Nth number with number divs*/
end /*s*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _; do c=length(_)-3 to 1 by -3; _=insert(',', _, c); end; return _
done: do f=N by -1 for N-3; if \zfin.f then return 0; end; return 1
pPow: numeric digits 2000; return @.j**(j-1) /*temporarily increase decimal digits. */
tell: parse arg _; say center(i,10) right(_,max(w,length(_))); if i//5==0 then say; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
#divs: procedure; parse arg x 1 y /*X and Y: both set from 1st argument.*/
if x<7 then do /*handle special cases for numbers < 7.*/
if x<3 then return x /* " " " " one and two.*/
if x<5 then return x - 1 /* " " " " three & four*/
if x==5 then return 2 /* " " " " five. */
if x==6 then return 4 /* " " " " six. */
end
odd= x // 2 /*check if X is odd or not. */
if odd then do; #= 1; end /*Odd? Assume Pdivisors count of 1.*/
else do; #= 3; y= x%2; end /*Even? " " " " 3.*/
/* [↑] start with known num of Pdivs.*/
do k=3 by 1+odd while k<y /*when doing odd numbers, skip evens. */
if x//k==0 then do /*if no remainder, then found a divisor*/
#=#+2; y=x%k /*bump # Pdivs, calculate limit Y. */
if k>=y then do; #= #-1; leave; end /*limit?*/
end /* ___ */
else if k*k>x then leave /*only divide up to √ x */
end /*k*/ /* [↑] this form of DO loop is faster.*/
return #+1 /*bump "proper divisors" to "divisors".*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
isPrime: procedure; parse arg # . '' -1 _
if #<31 then do; if wordpos(#, '2 3 5 7 11 13 17 19 23 29')\==0 then return 1
if #<2 then return 0
end
if #// 2==0 then return 0; if #// 3==0 then return 0; if _==5 then return 0
if #// 7==0 then return 0; if #//11==0 then return 0; if #//11==0 then return 0
if #//13==0 then return 0; if #//17==0 then return 0; if #//19==0 then return 0
do i=23 by 6 until i*i>#; if #// i ==0 then return 0
if #//(i+2)==0 then return 0
end /*i*/ /* ___ */
return 1 /*Exceeded √ #  ? Then # is prime. */
output   is identical to the 1st REXX version.


Ring[edit]

 
load "stdlib.ring"
 
num = 0
limit = 22563490300366186081
 
see "working..." + nl
see "the first 15 terms of the sequence are:" + nl
 
for n = 1 to 15
num = 0
for m = 1 to limit
pnum = 0
for p = 1 to limit
if (m % p = 0)
pnum = pnum + 1
ok
next
if pnum = n
num = num + 1
if num = n
see "" + n + ": " + m + " " + nl
exit
ok
ok
next
next
 
see nl + "done..." + nl
 
Output:
working...
the first 15 terms of the sequence are:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14641
 6 : 44
 7 : 24137569
 8 : 70
 9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
done...

Ruby[edit]

Translation of: Java
def isPrime(n)
return false if n < 2
return n == 2 if n % 2 == 0
return n == 3 if n % 3 == 0
 
k = 5
while k * k <= n
return false if n % k == 0
k = k + 2
end
 
return true
end
 
def getSmallPrimes(numPrimes)
smallPrimes = [2]
count = 0
n = 3
while count < numPrimes
if isPrime(n) then
smallPrimes << n
count = count + 1
end
n = n + 2
end
return smallPrimes
end
 
def getDivisorCount(n)
count = 1
while n % 2 == 0
n = (n / 2).floor
count = count + 1
end
 
d = 3
while d * d <= n
q = (n / d).floor
r = n % d
dc = 0
while r == 0
dc = dc + count
n = q
q = (n / d).floor
r = n % d
end
count = count + dc
d = d + 2
end
if n != 1 then
count = 2 * count
end
return count
end
 
MAX = 15
@smallPrimes = getSmallPrimes(MAX)
 
def OEISA073916(n)
if isPrime(n) then
return @smallPrimes[n - 1] ** (n - 1)
end
 
count = 0
result = 0
i = 1
while count < n
if n % 2 == 1 then
# The solution for an odd (non-prime) term is always a square number
root = Math.sqrt(i)
if root * root != i then
i = i + 1
next
end
end
if getDivisorCount(i) == n then
count = count + 1
result = i
end
i = i + 1
end
return result
end
 
n
= 1
while n <= MAX
print "A073916(", n, ") = ", OEISA073916(n), "\n"
n = n + 1
end
Output:
A073916(1) = 1
A073916(2) = 3
A073916(3) = 25
A073916(4) = 14
A073916(5) = 14641
A073916(6) = 44
A073916(7) = 24137569
A073916(8) = 70
A073916(9) = 1089
A073916(10) = 405
A073916(11) = 819628286980801
A073916(12) = 160
A073916(13) = 22563490300366186081
A073916(14) = 2752
A073916(15) = 9801

Sidef[edit]

func f(n {.is_prime}) {
n.prime**(n-1)
}
 
func f(n) {
n.th { .sigma0 == n }
}
 
say 20.of { f(_+1) }
Output:
[1, 3, 25, 14, 14641, 44, 24137569, 70, 1089, 405, 819628286980801, 160, 22563490300366186081, 2752, 9801, 462, 21559177407076402401757871041, 1044, 740195513856780056217081017732809, 1520]

Wren[edit]

Translation of: Kotlin
Library: Wren-math
Library: Wren-big
Library: Wren-fmt
import "/math" for Int
import "/big" for BigInt
import "/fmt" for Fmt
 
var MAX = 33
var primes = Int.primeSieve(MAX * 5)
System.print("The first %(MAX) terms in the sequence are:")
for (i in 1..MAX) {
if (Int.isPrime(i)) {
var z = BigInt.new(primes[i-1]).pow(i-1)
Fmt.print("$2d : $i", i, z)
} else {
var count = 0
var j = 1
while (true) {
var cont = false
if (i % 2 == 1) {
var sq = j.sqrt.floor
if (sq * sq != j) {
j = j + 1
cont = true
}
}
if (!cont) {
if (Int.divisors(j).count == i) {
count = count + 1
if (count == i) {
Fmt.print("$2d : $d", i, j)
break
}
}
j = j + 1
}
}
}
}
Output:
The first 33 terms in the sequence are:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14641
 6 : 44
 7 : 24137569
 8 : 70
 9 : 1089
10 : 405
11 : 819628286980801
12 : 160
13 : 22563490300366186081
14 : 2752
15 : 9801
16 : 462
17 : 21559177407076402401757871041
18 : 1044
19 : 740195513856780056217081017732809
20 : 1520
21 : 141376
22 : 84992
23 : 1658509762573818415340429240403156732495289
24 : 1170
25 : 52200625
26 : 421888
27 : 52900
28 : 9152
29 : 1116713952456127112240969687448211536647543601817400964721
30 : 6768
31 : 1300503809464370725741704158412711229899345159119325157292552449
32 : 3990
33 : 12166144

zkl[edit]

Translation of: Go

Using GMP (GNU Multiple Precision Arithmetic Library, probabilistic primes), because it is easy and fast to generate primes.

Extensible prime generator#zkl could be used instead.

var [const] BI=Import("zklBigNum"), pmax=25;  // libGMP
p:=BI(1);
primes:=pmax.pump(List(0), p.nextPrime, "copy"); //-->(0,3,5,7,11,13,17,19,...)
 
fcn countDivisors(n){
count:=1;
while(n%2==0){ n/=2; count+=1; }
foreach d in ([3..*,2]){
q,r := n/d, n%d;
if(r==0){
dc:=0;
while(r==0){
dc+=count;
n,q,r = q, n/d, n%d;
}
count+=dc;
}
if(d*d > n) break;
}
if(n!=1) count*=2;
count
}
 
println("The first ", pmax, " terms in the sequence are:");
foreach i in ([1..pmax]){
if(BI(i).probablyPrime()) println("%2d : %,d".fmt(i,primes[i].pow(i-1)));
else{
count:=0;
foreach j in ([1..*]){
if(i%2==1 and j != j.toFloat().sqrt().toInt().pow(2)) continue;
if(countDivisors(j) == i){
count+=1;
if(count==i){
println("%2d : %,d".fmt(i,j));
break;
}
}
}
}
}
Output:
The first 25 terms in the sequence are:
 1 : 1
 2 : 3
 3 : 25
 4 : 14
 5 : 14,641
 6 : 44
 7 : 24,137,569
 8 : 70
 9 : 1,089
10 : 405
11 : 819,628,286,980,801
12 : 160
13 : 22,563,490,300,366,186,081
14 : 2,752
15 : 9,801
16 : 462
17 : 21,559,177,407,076,402,401,757,871,041
18 : 1,044
19 : 740,195,513,856,780,056,217,081,017,732,809
20 : 1,520
21 : 141,376
22 : 84,992
23 : 1,658,509,762,573,818,415,340,429,240,403,156,732,495,289
24 : 1,170
25 : 52,200,625