From 5029ecbca72583e46fe03363fcef798e078b7828 Mon Sep 17 00:00:00 2001
From: trizen <trizen@protonmail.com>
Date: Sun, 3 Nov 2019 20:36:36 +0200
Subject: [PATCH] - Added the Number `nth_composite(n)` and
 `composite_count(n)` methods.

For computing the n-th composite number (A002808), and, respectively, the number of composite numbers <= n (A065855).

Example:

	say nth_composite(10**9)		#=> 1053422339
	say composite_count(1053422339)		#=> 1000000000
---
 lib/Sidef/Types/Number/Number.pm  | 57 +++++++++++++++++++++++++++++++
 lib/Sidef/Types/Number/Number.pod | 30 +++++++++++++---
 2 files changed, 83 insertions(+), 4 deletions(-)

diff --git a/lib/Sidef/Types/Number/Number.pm b/lib/Sidef/Types/Number/Number.pm
index 86ca02add..48dadf799 100644
--- a/lib/Sidef/Types/Number/Number.pm
+++ b/lib/Sidef/Types/Number/Number.pm
@@ -8913,6 +8913,59 @@ package Sidef::Types::Number::Number {
 
     *prime = \&nth_prime;
 
+    sub composite_count {
+        my ($n) = @_;
+        $n->sub($n->prime_count)->dec;    # n - pi(n) - 1
+    }
+
+    sub nth_composite {
+        my ($n) = @_;
+        $n = _any2ui($$n) // goto &nan;
+
+        return ONE if ($n == 0);                         # not composite, but...
+        return __PACKAGE__->_set_uint(4) if ($n == 1);
+
+        # Lower and upper bounds from A002808 (for n >= 4).
+        my $min = CORE::int($n + $n / CORE::log($n) + $n / (CORE::log($n)**2));
+        my $max = CORE::int($n + $n / CORE::log($n) + (3 * $n) / (CORE::log($n)**2));
+
+        if ($n < 4) {
+            $min = 4;
+            $max = 8;
+        }
+
+        my $k = 0;
+
+        while (1) {
+            $k = ($min + $max) >> 1;
+
+            my $pi =
+              $HAS_PRIME_UTIL
+              ? Math::Prime::Util::prime_count($k)
+              : ${__PACKAGE__->_set_uint($k)->prime_count};
+
+            my $cmp = ($k <=> ($pi + 1 + $n));
+
+            if ($cmp > 0) {
+                $max = $k - 1;
+            }
+            elsif ($cmp < 0) {
+                $min = $k + 1;
+            }
+            else {
+                last;
+            }
+        }
+
+        if ($HAS_PRIME_UTIL ? Math::Prime::Util::is_prime($k) : Math::Prime::Util::GMP::is_prob_prime($k)) {
+            --$k;
+        }
+
+        __PACKAGE__->_set_uint($k);
+    }
+
+    *composite = \&nth_composite;
+
     sub legendre {
         my ($x, $y) = @_;
         _valid(\$y);
@@ -9556,6 +9609,8 @@ package Sidef::Types::Number::Number {
           : Sidef::Types::Bool::Bool::FALSE;
     }
 
+    *is_nm1_prime = \&is_nminus1_prime;
+
     sub is_nplus1_prime {
         my ($x) = @_;
 
@@ -9568,6 +9623,8 @@ package Sidef::Types::Number::Number {
           : Sidef::Types::Bool::Bool::FALSE;
     }
 
+    *is_np1_prime = \&is_nplus1_prime;
+
     sub is_ecpp_prime {
         my ($x) = @_;
 
diff --git a/lib/Sidef/Types/Number/Number.pod b/lib/Sidef/Types/Number/Number.pod
index edabaae0e..c7b0ba0b5 100644
--- a/lib/Sidef/Types/Number/Number.pod
+++ b/lib/Sidef/Types/Number/Number.pod
@@ -1130,6 +1130,24 @@ Return the
 
 =cut
 
+=head2 composite
+
+Number.composite() -> I<Obj>
+
+Return the
+
+Aliases: I<nth_composite>
+
+=cut
+
+=head2 composite_count
+
+Number.composite_count() -> I<Obj>
+
+Return the
+
+=cut
+
 =head2 conj
 
 Number.conj() -> I<Obj>
@@ -2172,20 +2190,24 @@ Return the
 
 =cut
 
-=head2 is_nminus1_prime
+=head2 is_nm1_prime
 
-Number.is_nminus1_prime() -> I<Obj>
+Number.is_nm1_prime() -> I<Obj>
 
 Return the
 
+Aliases: I<is_nminus1_prime>
+
 =cut
 
-=head2 is_nplus1_prime
+=head2 is_np1_prime
 
-Number.is_nplus1_prime() -> I<Obj>
+Number.is_np1_prime() -> I<Obj>
 
 Return the
 
+Aliases: I<is_nplus1_prime>
+
 =cut
 
 =head2 is_odd