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{{
In a manner similar to twin primes, twin Ramanujan primes may be explored. The task is to determine how many of the first million Ramanujan primes are twins.
;Related Task: [[Twin primes]]
=={{header|C++}}==
<syntaxhighlight lang="c++">
#include <cmath>
#include <cstdint>
#include <iostream>
#include <vector>
int32_t ramanujan_maximum(const int32_t& number) {
return ceil(4 * number * log(4 * number));
}
std::vector<int32_t> initialise_prime_pi(const int32_t& limit) {
std::vector<int32_t> result(limit, 1);
result[0] = 0;
result[1] = 0;
for ( int32_t i = 4; i < limit; i += 2 ) {
result[i] = 0;
}
for ( int32_t p = 3, square = 9; square < limit; p += 2 ) {
if ( result[p] != 0 ) {
for ( int32_t q = square; q < limit; q += p << 1 ) {
result[q] = 0;
}
}
square += ( p + 1 ) << 2;
}
for ( uint64_t i = 1; i < result.size(); ++i ) {
result[i] += result[i - 1];
}
return result;
}
int32_t ramanujan_prime(const std::vector<int32_t>& prime_pi, const int32_t& number) {
int32_t maximum = ramanujan_maximum(number);
if ( ( maximum & 1) == 1 ) {
maximum -= 1;
}
int32_t index = maximum;
while ( prime_pi[index] - prime_pi[index / 2] >= number ) {
index -= 1;
}
return index + 1;
}
std::vector<int32_t> list_primes_less_than(const int32_t& limit) {
std::vector<bool> composite(limit, false);
int32_t n = 3;
int32_t nSquared = 9;
while ( nSquared <= limit ) {
if ( ! composite[n] ) {
for ( int32_t k = nSquared; k < limit; k += 2 * n ) {
composite[k] = true;
}
}
nSquared += ( n + 1 ) << 2;
n += 2;
}
std::vector<int32_t> result;
result.emplace_back(2);
for ( int32_t i = 3; i < limit; i += 2 ) {
if ( ! composite[i] ) {
result.emplace_back(i);
}
}
return result;
}
int main() {
const int32_t limit = 1'000'000;
const std::vector<int32_t> prime_pi = initialise_prime_pi(ramanujan_maximum(limit) + 1);
const int32_t millionth_ramanujan_prime = ramanujan_prime(prime_pi, limit);
std::cout << "The 1_000_000th Ramanujan prime is " << millionth_ramanujan_prime << std::endl;
std::vector<int32_t> primes = list_primes_less_than(millionth_ramanujan_prime);
std::vector<int32_t> ramanujan_prime_indexes(primes.size());
for ( uint64_t i = 0; i < ramanujan_prime_indexes.size(); ++i ) {
ramanujan_prime_indexes[i] = prime_pi[primes[i]] - prime_pi[primes[i] / 2];
}
int32_t lowerLimit = ramanujan_prime_indexes[ramanujan_prime_indexes.size() - 1];
for ( int32_t i = ramanujan_prime_indexes.size() - 2; i >= 0; --i ) {
if ( ramanujan_prime_indexes[i] < lowerLimit ) {
lowerLimit = ramanujan_prime_indexes[i];
} else {
ramanujan_prime_indexes[i] = 0;
}
}
std::vector<int32_t> ramanujan_primes;
for ( uint64_t i = 0; i < ramanujan_prime_indexes.size(); ++i ) {
if ( ramanujan_prime_indexes[i] != 0 ) {
ramanujan_primes.emplace_back(primes[i]);
}
}
int32_t twins_count = 0;
for ( uint64_t i = 0; i < ramanujan_primes.size() - 1; ++i ) {
if ( ramanujan_primes[i] + 2 == ramanujan_primes[i + 1] ) {
twins_count++;
}
}
std::cout << "There are " << twins_count << " twins in the first " << limit << " Ramanujan primes." << std::endl;
}
</syntaxhighlight>
{{ out }}
<pre>
The 1_000_000th Ramanujan prime is 34072993
There are 74973 twins in the first 1000000 Ramanujan primes.
</pre>
=={{header|EasyLang}}==
<syntaxhighlight>
global cnt[] .
proc primcnt limit . .
cnt[] = [ 0 1 1 ]
for i = 4 step 2 to limit
cnt[] &= 0
cnt[] &= 1
.
p = 3
sq = 9
while sq <= limit
if cnt[p] <> 0
for q = sq step p * 2 to limit
cnt[q] = 0
.
.
sq += (p + 1) * 4
p += 2
.
for i = 2 to limit
sum += cnt[i]
cnt[i] = sum
.
.
func log n .
e = 2.7182818284590452354
return log10 n / log10 e
.
func ramamax n .
return floor (4 * n * log (4 * n))
.
func ramaprim_twins n .
i = ramamax n
i -= i mod 2
while i >= 2
if cnt[i] - cnt[i / 2] < n
p1 = p
p = i + 1
if p1 - p = 2
cnt += 1
.
n -= 1
.
i -= 2
.
return cnt
.
primcnt ramamax 1000000
print ramaprim_twins 1000000
</syntaxhighlight>
{{out}}
<pre>
74973
</pre>
=={{header|F_Sharp|F#}}==
This task uses [http://www.rosettacode.org/wiki/Ramanujan_primes#F.23 Ramanujan primes (F#)]
<
// Twin Ramanujan primes. Nigel Galloway: September 9th., 2021
printfn $"There are %d{rP 1000000|>Seq.pairwise|>Seq.filter(fun(n,g)->n=g-2)|>Seq.length} twins in the first million Ramanujan primes"
</syntaxhighlight>
{{out}}
<pre>
There are 74973 twins in the first million Ramanujan primes
</pre>
=={{header|FreeBASIC}}==
<syntaxhighlight lang="vbnet">#define floor(x) ((x*2.0-0.5) Shr 1)
Dim Shared pi() As Integer
Sub primeCounter(limit As Integer)
Dim As Integer i, q, p, sq, total
Redim pi(limit)
pi(0) = 0
pi(1) = 0
For i = 2 To limit
pi(i) = 1
Next
If limit > 2 Then
For i = 4 To limit Step 2
pi(i) = 0
Next i
p = 3
sq = 9
While sq <= limit
If pi(p) <> 0 Then
For q = sq To limit Step p*2
pi(q) = 0
Next q
End If
sq += (p + 1) * 4
p += 2
Wend
total = 0
For i = 2 To limit
total += pi(i)
pi(i) = total
Next i
End If
End Sub
Function ramanujanMax(n As Integer) As Integer
Return floor(4 * n * Log(4*n))
End Function
Function ramanujanPrimeTwins(n As Integer) As Integer
Dim As Integer maxposs, p1, p, cnt
maxposs = ramanujanMax(n)
maxposs -= maxposs Mod 2
While maxposs >= 2
If pi(maxposs) - pi(maxposs \ 2) < n Then
p1 = p
p = maxposs + 1
If p1 - p = 2 Then cnt += 1
n -= 1
End If
maxposs -= 2
Wend
Return cnt
End Function
Dim As Integer limit = 1e6
Dim As Double t0 = Timer
primeCounter ramanujanMax(limit)
Print Using "There are & twins in the first & Ramanujan primes"; ramanujanPrimeTwins(limit); limit
Print Using "##.##sec."; Timer - t0
Sleep</syntaxhighlight>
{{out}}
<pre>There are 74973 twins in the first 1000000 Ramanujan primes
1.53sec.</pre>
=={{header|Go}}==
{{trans|Wren}}
{{libheader|Go-rcu}}
<syntaxhighlight lang="go">package main
import (
"fmt"
"math"
"rcu"
"time"
)
var count []int
func primeCounter(limit int) {
count = make([]int, limit)
for i := 0; i < limit; i++ {
count[i] = 1
}
if limit > 0 {
count[0] = 0
}
if limit > 1 {
count[1] = 0
}
for i := 4; i < limit; i += 2 {
count[i] = 0
}
for p, sq := 3, 9; sq < limit; p += 2 {
if count[p] != 0 {
for q := sq; q < limit; q += p << 1 {
count[q] = 0
}
}
sq += (p + 1) << 2
}
sum := 0
for i := 0; i < limit; i++ {
sum += count[i]
count[i] = sum
}
}
func primeCount(n int) int {
if n < 1 {
return 0
}
return count[n]
}
func ramanujanMax(n int) int {
fn := float64(n)
return int(math.Ceil(4 * fn * math.Log(4*fn)))
}
func ramanujanPrime(n int) int {
if n == 1 {
return 2
}
for i := ramanujanMax(n); i >= 2*n; i-- {
if i%2 == 1 {
continue
}
if primeCount(i)-primeCount(i/2) < n {
return i + 1
}
}
return 0
}
func rpc(p int) int { return primeCount(p) - primeCount(p/2) }
func main() {
for _, limit := range []int{1e5, 1e6} {
start := time.Now()
primeCounter(1 + ramanujanMax(limit))
rplim := ramanujanPrime(limit)
climit := rcu.Commatize(limit)
fmt.Printf("The %sth Ramanujan prime is %s\n", climit, rcu.Commatize(rplim))
r := rcu.Primes(rplim)
c := make([]int, len(r))
for i := 0; i < len(c); i++ {
c[i] = rpc(r[i])
}
ok := c[len(c)-1]
for i := len(c) - 2; i >= 0; i-- {
if c[i] < ok {
ok = c[i]
} else {
c[i] = 0
}
}
var fr []int
for i, r := range r {
if c[i] != 0 {
fr = append(fr, r)
}
}
twins := 0
for i := 0; i < len(fr)-1; i++ {
if fr[i]+2 == fr[i+1] {
twins++
}
}
fmt.Printf("There are %s twins in the first %s Ramanujan primes.\n", rcu.Commatize(twins), climit)
fmt.Println("Took", time.Since(start))
fmt.Println()
}
}</syntaxhighlight>
{{out}}
<pre>
The 100,000th Ramanujan prime is 2,916,539
There are 8,732 twins in the first 100,000 Ramanujan primes.
Took 50.745239ms
The 1,000,000th Ramanujan prime is 34,072,993
There are 74,973 twins in the first 1,000,000 Ramanujan primes.
Took 699.967994ms
</pre>
=={{header|J}}==
<syntaxhighlight lang="j"> _2 +/@:= 2 -/\ (((] - _1&(33 b.)@:>:@[ { ]) _1&p:) i. 35e6) i: i. 1e6
74973</syntaxhighlight>
=={{header|Java}}==
<syntaxhighlight lang="java">
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
public final class RamanujanPrimesTwins {
public static void main(String[] aArgs) {
final int limit = 1_000_000;
final int[] primePi = initialisePrimePi(ramanujanMaximum(limit) + 1);
final int millionthRamanujanPrime = ramanujanPrime(primePi, limit);
System.out.println("The 1_000_000th Ramanujan prime is " + millionthRamanujanPrime);
List<Integer> primes = listPrimesLessThan(millionthRamanujanPrime);
int[] ramanujanPrimeIndexes = new int[primes.size()];
for ( int i = 0; i < ramanujanPrimeIndexes.length; i++ ) {
ramanujanPrimeIndexes[i] = primePi[primes.get(i)] - primePi[primes.get(i) / 2];
}
int lowerLimit = ramanujanPrimeIndexes[ramanujanPrimeIndexes.length - 1];
for ( int i = ramanujanPrimeIndexes.length - 2; i >= 0; i-- ) {
if ( ramanujanPrimeIndexes[i] < lowerLimit ) {
lowerLimit = ramanujanPrimeIndexes[i];
} else {
ramanujanPrimeIndexes[i] = 0;
}
}
List<Integer> ramanujanPrimes = new ArrayList<Integer>();
for ( int i = 0; i < ramanujanPrimeIndexes.length; i++ ) {
if ( ramanujanPrimeIndexes[i] != 0 ) {
ramanujanPrimes.add(primes.get(i));
}
}
int twinsCount = 0;
for ( int i = 0; i < ramanujanPrimes.size() - 1; i++ ) {
if ( ramanujanPrimes.get(i) + 2 == ramanujanPrimes.get(i + 1) ) {
twinsCount += 1;
}
}
System.out.println("There are " + twinsCount + " twins in the first " + limit + " Ramanujan primes.");
}
private static List<Integer> listPrimesLessThan(int aLimit) {
boolean[] composite = new boolean[aLimit + 1];
int n = 3;
int nSquared = 9;
while ( nSquared <= aLimit ) {
if ( ! composite[n] ) {
for ( int k = nSquared; k < aLimit; k += 2 * n ) {
composite[k] = true;
}
}
nSquared += ( n + 1 ) << 2;
n += 2;
}
List<Integer> result = new ArrayList<Integer>();
result.add(2);
for ( int i = 3; i < aLimit; i += 2 ) {
if ( ! composite[i] ) {
result.add(i);
}
}
return result;
}
private static int ramanujanPrime(int[] aPrimePi, int aNumber) {
int maximum = ramanujanMaximum(aNumber);
if ( ( maximum & 1) == 1 ) {
maximum -= 1;
}
int index = maximum;
while ( aPrimePi[index] - aPrimePi[index / 2] >= aNumber ) {
index -= 1;
}
return index + 1;
}
private static int[] initialisePrimePi(int aLimit) {
int[] result = new int[aLimit];
Arrays.fill(result, 1);
result[0] = 0;
result[1] = 0;
for ( int i = 4; i < aLimit; i += 2 ) {
result[i] = 0;
}
for ( int p = 3, square = 9; square < aLimit; p += 2 ) {
if ( result[p] != 0 ) {
for ( int q = square; q < aLimit; q += p << 1 ) {
result[q] = 0;
}
}
square += ( p + 1 ) << 2;
}
Arrays.parallelPrefix(result, Integer::sum);
return result;
}
private static int ramanujanMaximum(int aNumber) {
return (int) Math.ceil(4 * aNumber * Math.log(4 * aNumber));
}
}
</syntaxhighlight>
{{ out }}
<pre>
The 1_000_000th Ramanujan prime is 34072993
There are 74973 twins in the first 1000000 Ramanujan primes.
</pre>
=={{header|Julia}}==
<syntaxhighlight lang="julia">using Primes
function rajpairs(N, verbose, interval = 2)
maxpossramanujan(n) = Int(ceil(4n * log(4n) / log(2)))
prm = primes(maxpossramanujan(N))
pivec = accumulate(+, primesmask(maxpossramanujan(N)))
halfrpc = [pivec[p] - pivec[p ÷ 2] for p in prm]
lastrpc = last(halfrpc) + 1
for i in length(halfrpc):-1:1
if halfrpc[i] < lastrpc
lastrpc = halfrpc[i]
else
halfrpc[i] = 0
end
end
rajvec = [prm[i] for i in eachindex(prm) if halfrpc[i] > 0]
nrajtwins = count(rajvec[i] + interval == rajvec[i + 1] for i in 1:N-1)
verbose && println("There are $nrajtwins twins in the first $N Ramanujan primes.")
return nrajtwins
end
rajpairs(100000, false)
@time rajpairs(1000000, true)
</syntaxhighlight>{{out}}
<pre>
There are 74973 twins in the first 1000000 Ramanujan primes.
0.759511 seconds (50 allocations: 839.383 MiB, 5.64% gc time)
</pre>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">$HistoryLength = 1;
l = PrimePi[Range[35 10^6]] - PrimePi[Range[35 10^6]/2];
ramanujanprimes = GatherBy[Transpose[{Range[2, Length[l] + 1], l}], Last][[All, -1, 1]];
ramanujanprimes = Take[Sort@ramanujanprimes, 10^6];
Count[Differences[ramanujanprimes], 2]</syntaxhighlight>
{{out}}
<pre>74973</pre>
=={{header|Nim}}==
{{trans|Go}}
We use Phix algorithm in its Go translation.
<syntaxhighlight lang="nim">import math, sequtils, strutils, sugar, times
let t0 = now()
type PrimeCounter = seq[int32]
proc initPrimeCounter(limit: Positive): PrimeCounter {.noInit.} =
doAssert limit > 1
result = repeat(1i32, limit)
result[0] = 0
result[1] = 0
for i in countup(4, limit - 1, 2): result[i] = 0
var p = 3
var p2 = 9
while p2 < limit:
if result[p] != 0:
for q in countup(p2, limit - 1, p + p):
result[q] = 0
inc p, 2
p2 = p * p
# Compute partial sums in place.
var sum = 0i32
for item in result.mitems:
sum += item
item = sum
func ramanujanMax(n: int): int {.inline.} = int(ceil(4 * n.toFloat * ln(4 * n.toFloat)))
func ramanujanPrime(pi: PrimeCounter; n: int): int =
if n == 1: return 2
var max = ramanujanMax(n)
if (max and 1) == 1: dec max
for i in countdown(max, 2, 2):
if pi[i] - pi[i div 2] < n:
return i + 1
func primesLe(limit: Positive): seq[int] =
var composite = newSeq[bool](limit + 1)
var n = 3
var n2 = 9
while n2 <= limit:
if not composite[n]:
for k in countup(n2, limit, 2 * n):
composite[k] = true
n2 += (n + 1) shl 2
n += 2
result = @[2]
for n in countup(3, limit, 2):
if not composite[n]: result.add n
proc main() =
const Lim = 1_000_000
let pi = initPrimeCounter(1 + ramanujanMax(Lim))
let rpLim = ramanujanPrime(pi, Lim)
echo "The 1_000_000th Ramanujan prime is $#.".format(($rpLim).insertSep())
let r = primesLe(rpLim)
var c = r.mapIt(pi[it] - pi[it div 2])
var ok = c[^1]
for i in countdown(c.len - 2, 0):
if c[i] < ok:
ok = c[i]
else:
c[i] = 0
let fr = collect(newSeq, for i, p in r: (if c[i] != 0: p))
var twins = 0
var prev = -1
for p in fr:
if p == prev + 2: inc twins
prev = p
echo "There are $1 twins in the first $2 Ramanujan primes.".format(($twins).insertSep(), ($Lim).insertSep)
main()
echo "\nElapsed time: ", (now() - t0).inMilliseconds, " ms"</syntaxhighlight>
{{out}}
<pre>The 1_000_000th Ramanujan prime is 34_072_993.
There are 74_973 twins in the first 1_000_000 Ramanujan primes.
Elapsed time: 1139 ms</pre>
=={{header|Perl}}==
Can't do better than the <code>ntheory</code> module.
{{libheader|ntheory}}
<syntaxhighlight lang="perl">use strict;
use warnings;
use ntheory <ramanujan_primes nth_ramanujan_prime>;
sub comma { reverse ((reverse shift) =~ s/(.{3})/$1,/gr) =~ s/^,//r }
my $rp = ramanujan_primes nth_ramanujan_prime 1_000_000;
for my $limit (1e5, 1e6) {
printf "The %sth Ramanujan prime is %s\n", comma($limit), comma $rp->[$limit-1];
printf "There are %s twins in the first %s Ramanujan primes.\n\n",
comma( scalar grep { $rp->[$_+1] == $rp->[$_]+2 } 0 .. $limit-2 ), comma $limit;
}</syntaxhighlight>
{{out}}
<pre>The 100,000th Ramanujan prime is 2,916,539
There are 8,732 twins in the first 100,000 Ramanujan primes.
The 1,000,000th Ramanujan prime is 34,072,993
There are 74,973 twins in the first 1,000,000 Ramanujan primes.</pre>
=={{header|Phix}}==
While finding the 1,000,000th Ramanujan prime is reasonably cheap (~
trick to find all 1,000,000 of them individually is over
Calculating pi(p) - pi(floor(pi/2) for all (normal) primes below that one millionth, and then
filtering them based on that list is significantly faster, and finally we can scan for twins.
<!--<
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #
<span style="color: #
<span style="color: #008080;">
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #
<span style="color: #008080;">for</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">=</span><span style="color: #000000;">sq</span> <span style="color: #008080;">to</span> <span style="color: #000000;">limit</span> <span style="color: #008080;">by</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">pi</span><span style="color: #0000FF;">[</span><span style="color: #000000;">q</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">sq</span> <span style="color: #0000FF;">+=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">p</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">4</span>
<span style="color: #000000;">p</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">2</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">total</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">limit</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">total</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">pi</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">pi</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">total</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">ramanujanMax</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">n</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">n</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">
<span style="color: #
<span style="color: #
<span style="color: #008080;">if</span> <span style="color: #000000;">pi</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">pi</span><span style="color: #0000FF;">[</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)]</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">n</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">i</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
Line 47 ⟶ 684:
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #
--constant lim = 1e6 -- 4.7s -- (not pwa/p2js)</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #000000;">primeCounter</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ramanujanMax</span><span style="color: #0000FF;">(</span><span style="color: #000000;">lim</span><span style="color: #0000FF;">))</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">rplim</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ramanujanPrime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">lim</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"The %,dth Ramanujan prime is %,d\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">lim</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rplim</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">rpc</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #000000;">pi</span><span style="color: #0000FF;">
<span style="color: #004080;">sequence</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">get_primes_le</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rplim</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">apply</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rpc</span><span style="color: #0000FF;">)</span>
Line 68 ⟶ 709:
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"There are %,d twins in the first %,d Ramanujan primes\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">twins</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)})</span>
<span style="color: #0000FF;">?</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<!--</
{{out}}
using a smaller limit:
Line 74 ⟶ 715:
The 100,000th Ramanujan prime is 2,916,539
There are 8,732 twins in the first 100,000 Ramanujan primes
"
</pre>
with the higher limit (desktop/Phix only, alas not possible under pwa/p2js):
<pre>
The 1,000,000th Ramanujan prime is 34,072,993
There are 74,973 twins in the first 1,000,000 Ramanujan primes
"4.7s"
</pre>
=={{header|Raku}}==
Timing is informational only. Will vary by system specs and load.
{{libheader|ntheory}}
<syntaxhighlight lang="raku" line>use ntheory:from<Perl5> <ramanujan_primes nth_ramanujan_prime>;
use Lingua::EN::Numbers;
my @rp = ramanujan_primes nth_ramanujan_prime 1_000_000;
for (1e5, 1e6)».Int -> $limit {
say "\nThe {comma $limit}th Ramanujan prime is { comma @rp[$limit-1]}";
say "There are { comma +(^($limit-1)).race.grep: { @rp[$_+1] == @rp[$_]+2 } } twins in the first {comma $limit} Ramanujan primes.";
}
say (now - INIT now).fmt('%.3f') ~ ' seconds';</syntaxhighlight>
{{out}}
<pre>The 100,000th Ramanujan prime is 2,916,539
There are 8,732 twins in the first 100,000 Ramanujan primes.
The 1,000,000th Ramanujan prime is 34,072,993
There are 74,973 twins in the first 1,000,000 Ramanujan primes.
2.529 seconds</pre>
=={{header|Scala}}==
{{trans|Java}}
<syntaxhighlight lang="Scala">
import java.util.{ArrayList, Arrays, List}
object RamanujanPrimesTwins {
def main(args: Array[String]): Unit = {
val limit = 1_000_000
val primePi = initialisePrimePi(ramanujanMaximum(limit) + 1)
val millionthRamanujanPrime = ramanujanPrime(primePi, limit)
println(s"The 1_000_000th Ramanujan prime is $millionthRamanujanPrime")
val primes = listPrimesLessThan(millionthRamanujanPrime)
val ramanujanPrimeIndexes = new Array[Int](primes.size)
for (i <- 0 until primes.size by 1) {
ramanujanPrimeIndexes(i) = primePi(primes.get(i)) - primePi(primes.get(i) / 2)
}
var lowerLimit = ramanujanPrimeIndexes(ramanujanPrimeIndexes.length - 1)
for (i <- ramanujanPrimeIndexes.length - 2 to 0 by -1) {
if (ramanujanPrimeIndexes(i) < lowerLimit) {
lowerLimit = ramanujanPrimeIndexes(i)
} else {
ramanujanPrimeIndexes(i) = 0
}
}
val ramanujanPrimes = new ArrayList[Integer]()
for (i <- 0 until ramanujanPrimeIndexes.size by 1) {
if (ramanujanPrimeIndexes(i) != 0) {
ramanujanPrimes.add(primes.get(i))
}
}
var twinsCount = 0
for (i <- 0 until ramanujanPrimes.size - 1) {
if (ramanujanPrimes.get(i) + 2 == ramanujanPrimes.get(i + 1)) {
twinsCount += 1
}
}
println(s"There are $twinsCount twins in the first $limit Ramanujan primes.")
}
private def listPrimesLessThan(limit: Int): List[Integer] = {
val composite = new Array[Boolean](limit + 1)
var n = 3
var nSquared = 9
while (nSquared <= limit) {
if (!composite(n)) {
for (k <- nSquared until limit by (2*n) ) {
composite(k) = true
}
}
nSquared += (n + 1) << 2
n += 2
}
var result = new ArrayList[Integer]()
result.add(2)
for (i <- 3 until limit by 2) {
if (!composite(i)) {
result.add(i)
}
}
result
}
private def ramanujanPrime(primePi: Array[Int], number: Int): Int = {
var maximum = ramanujanMaximum(number)
if ((maximum & 1) == 1) {
maximum -= 1
}
var index = maximum
while (primePi(index) - primePi(index / 2) >= number) {
index -= 1
}
index + 1
}
private def initialisePrimePi(aLimit : Int): Array[Int] = {
val result = new Array[Int](aLimit )
Arrays.fill(result, 1)
result(0) = 0
result(1) = 0
for (i <- 4 until aLimit by 2) {
result(i) = 0
}
var p = 3;var square = 9;
while(square < aLimit) {
if ( result(p) != 0 ) {
for ( q <- square until aLimit by (p << 1) ) {
result(q) = 0;
}
}
square += ( p + 1 ) << 2;
p += 2
}
for (i <- 1 until result.length) {
result(i) += result(i - 1)
}
result
}
private def ramanujanMaximum(number: Int): Int = {
Math.ceil(4 * number * Math.log(4 * number)).toInt
}
}
</syntaxhighlight>
{{out}}
<pre>
The 1_000_000th Ramanujan prime is 34072993
There are 74973 twins in the first 1000000 Ramanujan primes.
</pre>
=={{header|Sidef}}==
{{libheader|ntheory}}
<syntaxhighlight lang="ruby" line>require('ntheory')
var rp = %S<ntheory>.ramanujan_primes(%S<ntheory>.nth_ramanujan_prime(1e6))
for limit in ([1e5, 1e6]) {
printf("The %sth Ramanujan prime is %s\n", limit.commify, rp[limit-1].commify)
printf("There are %s twins in the first %s Ramanujan primes.\n\n",
^(limit-1) -> count {|i| rp[i+1] == rp[i]+2 }.commify, limit.commify)
}</syntaxhighlight>
{{out}}
<pre>The 100,000th Ramanujan prime is 2,916,539
There are 8,732 twins in the first 100,000 Ramanujan primes.
The 1,000,000th Ramanujan prime is 34,072,993
There are 74,973 twins in the first 1,000,000 Ramanujan primes.</pre>
=={{header|Wren}}==
{{trans|Phix}}
{{libheader|Wren-iterate}}
{{libheader|Wren-math}}
{{libheader|Wren-fmt}}
<br>
As calculating the first 1 million Ramanujan primes individually is out of the question, we follow the Phix entry's clever strategy here which brings this task comfortably within our ambit.
<syntaxhighlight lang="wren">import "./iterate" for Stepped, Indexed
import "./math" for Int, Math
import "./fmt" for Fmt
var count
var primeCounter = Fn.new { |limit|
count = List.filled(limit, 1)
if (limit > 0) count[0] = 0
if (limit > 1) count[1] = 0
for (i in Stepped.new(4...limit, 2)) count[i] = 0
var p = 3
var sq = 9
while (sq < limit) {
if (count[p] != 0) {
var q = sq
while (q < limit) {
count[q] = 0
q = q + p * 2
}
}
sq = sq + (p + 1) * 4
p = p + 2
}
var sum = 0
for (i in 0...limit) {
sum = sum + count[i]
count[i] = sum
}
}
var primeCount = Fn.new { |n| (n < 1) ? 0 : count[n] }
var ramanujanMax = Fn.new { |n| (4 * n * (4*n).log).ceil }
var ramanujanPrime = Fn.new { |n|
if (n == 1) return 2
for (i in ramanujanMax.call(n)..2) {
if (i % 2 == 1) continue
if (primeCount.call(i) - primeCount.call((i/2).floor) < n) return i + 1
}
return 0
}
var rpc = Fn.new { |p| primeCount.call(p) - primeCount.call((p/2).floor) }
for (limit in [1e5, 1e6]) {
var start = System.clock
primeCounter.call(1 + ramanujanMax.call(limit))
var rplim = ramanujanPrime.call(limit)
Fmt.print("The $,dth Ramanujan prime is $,d", limit, rplim)
var r = Int.primeSieve(rplim)
var c = r.map { |p| rpc.call(p) }.toList
var ok = c[-1]
for (i in c.count - 2..0) {
if (c[i] < ok) {
ok = c[i]
} else {
c[i] = 0
}
}
var ir = Indexed.new(r).where { |se| c[se.index] != 0 }.toList
var twins = 0
for (i in 0...ir.count-1) {
if (ir[i].value + 2 == ir[i+1].value) twins = twins + 1
}
Fmt.print("There are $,d twins in the first $,d Ramanujan primes.", twins, limit)
System.print("Took %(Math.toPlaces(System.clock - start, 2, 0)) seconds.\n")
}</syntaxhighlight>
{{out}}
<pre>
The 100,000th Ramanujan prime is 2,916,539
There are 8,732 twins in the first 100,000 Ramanujan primes.
Took 1.33 seconds.
The 1,000,000th Ramanujan prime is 34,072,993
There are 74,973 twins in the first 1,000,000 Ramanujan primes.
Took 15.75 seconds.
</pre>
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