Typo fix: "bellow" -> "below"

Author: trizen

Math/chernick-carmichael_numbers.sf

@@ -4,7 +4,7 @@
 # Date: 30 March 2019
 # https://github.com/trizen
 
-# Generate the extended Chernick-Carmichael numbers bellow a given limit.
+# Generate the extended Chernick-Carmichael numbers below a given limit.
 
 # See also:
 #   https://oeis.org/A317126
@@ -21,7 +21,7 @@ func is_chernick_carmichael (n, m) {
            : (is_prime(2**(n-2) * 9*m + 1) && __FUNC__(n-1, m))
 }
 
-# Generate the Chernick-Carmichael bellow a given limit
+# Generate the Chernick-Carmichael below a given limit
 func chernick_carmichael_numbers(limit) {
 
     var terms = []

Math/fermat_strong_primality_test.sf

@@ -38,9 +38,9 @@ func is_strong_fermat_pseudoprime(n, base=2) {
     return false
 }
 
-say ("Primes bellow 100: ", is_strong_fermat_pseudoprime.grep(1..100))
+say ("Primes below 100: ", is_strong_fermat_pseudoprime.grep(1..100))
 say ("First 5 strong pseudoprimes to base 2: ", { !.is_prime && is_strong_fermat_pseudoprime(_, 2) }.first(5))
 
 __END__
-Primes bellow 100: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
+Primes below 100: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
 First 5 strong pseudoprimes to base 2: [2047, 3277, 4033, 4681, 8321]

Math/hamming_numbers.sf

@@ -1,6 +1,6 @@
 #!/usr/bin/ruby
 
-# Generate the generalized Hamming numbers bellow a certain limit, given a set of primes.
+# Generate the generalized Hamming numbers below a certain limit, given a set of primes.
 
 func hamming_numbers (limit, primes) {
 

Math/left-right_truncatable_primes.sf

@@ -13,7 +13,7 @@
 # 2. Removing both digits, the number is still prime.
 # 3. Continuing this process, we end up with X.
 
-# Full sequence for prime numbers X bellow 10:
+# Full sequence for prime numbers X below 10:
 #   2, 3, 5, 7, 131, 137, 173, 179, 373, 379, 431, 479, 673, 971, 21319, 33739, 54799, 63793, 66733, 76733, 91373, 91733, 94793, 2913739, 3667333, 9637937, 696379373, 896379373
 
 # See also:

Math/modular_tetration.sf

@@ -1,6 +1,6 @@
 #!/usr/bin/ruby
 
-# Modular computation of the tetration operation, using using Euler's theorem.
+# Modular computation of the tetration operation, using Euler's theorem.
 
 # See also:
 #   https://en.wikipedia.org/wiki/Tetration

Math/primality_testing_fermat_fourier.sf

@@ -44,7 +44,7 @@ func is_fermat_prime_3(n) {
     (i*n*acoth(tanh((log(1 - exp(-((i*π*exp(log(2)*n))/n))) - log(1 - exp((i*π*exp(log(2)*n))/n)))/2)))/π
 }
 
-# Display the primes bellow 100
+# Display the primes below 100
 say (1..100 -> grep { is_fermat_prime_1(_) =~= 1 })
 say (1..100 -> grep { is_fermat_prime_2(_) =~= 1 })
 say (1..100 -> grep { is_fermat_prime_3(_) =~= 1 })

Math/primality_testing_wilson_fourier.sf

@@ -28,6 +28,6 @@ func is_wilson_prime_2(n) {
     (n*(i*log(1 - exp(-(2*i*π*Γ(n))/n)) - i*log(1 - exp((2*i*π*Γ(n))/n)) + π))/(2*π*(n-1))
 }
 
-# Display the primes bellow 100
+# Display the primes below 100
 say (1..100 -> grep { is_wilson_prime_1(_) =~= 1 })
 say (1..100 -> grep { is_wilson_prime_2(_) =~= 1 })

Math/semiprime_count.sf

@@ -1,23 +1,36 @@
 #!/usr/bin/ruby
 
-# Count the number of semiprimes bellow a certain limit.
+# Several algorithms for counting the number of semiprimes <= n.
 
 # See also:
 #   https://en.wikipedia.org/wiki/Semiprime
 
 func semiprime_count_1(n) {
     var t = 0
     n.isqrt.primes.sum {|p|
-        prime_count(n // p) - ++t + 1
+        prime_count(idiv(n,p)) - ++t + 1
     }
 }
 
 func semiprime_count_2(n) {
     n.isqrt.primes.sum {|p|
-        prime_count(p, n // p)
+        prime_count(p, idiv(n,p))
     }
 }
 
+func semiprime_count_3(n) {
+
+    var t = 2*n.isqrt.primes.sum {|p|
+        prime_count(idiv(n,p))
+    }
+
+    var r = prime_count(n.isqrt)
+
+    (t + r - r**2)/2
+}
+
 var n = irand(10**8)
+
 say semiprime_count_1(n)
 say semiprime_count_2(n)
+say semiprime_count_3(n)