Data commit

Task/Tau-function/Python/tau-function-1.py

@@ -0,0 +1,33 @@
+def factorize(n):
+    assert(isinstance(n, int))
+    if n < 0:
+        n = -n
+    if n < 2:
+        return
+    k = 0
+    while 0 == n%2:
+        k += 1
+        n //= 2
+    if 0 < k:
+        yield (2,k)
+    p = 3
+    while p*p <= n:
+        k = 0
+        while 0 == n%p:
+            k += 1
+            n //= p
+        if 0 < k:
+            yield (p,k)
+        p += 2
+    if 1 < n:
+        yield (n,1)
+
+def tau(n):
+    assert(n != 0)
+    ans = 1
+    for (p,k) in factorize(n):
+        ans *= 1 + k
+    return ans
+
+if __name__ == "__main__":
+    print(*map(tau, range(1, 101)))

Task/Tau-function/Python/tau-function-2.py

@@ -0,0 +1,17 @@
+def tau(n):
+    assert(isinstance(n, int) and 0 < n)
+    t = (n - 1 ^ n).bit_length()
+    n >>= t - 1
+    p = 3
+    while p * p <= n:
+        a = t
+        while n % p == 0:
+            t += a
+            n //= p
+        p += 2
+    if n != 1:
+        t += t
+    return t
+
+if __name__ == "__main__":
+    print(*map(tau, range(1, 101)))

Task/Tau-function/Python/tau-function-3.py

@@ -0,0 +1,87 @@
+'''The number of divisors of n'''
+
+from itertools import count, islice
+from math import floor, sqrt
+
+
+# oeisA000005 :: [Int]
+def oeisA000005():
+    '''tau(n) (also called d(n) or sigma_0(n)),
+       the number of divisors of n.
+    '''
+    return map(tau, count(1))
+
+
+# tau :: Int -> Int
+def tau(n):
+    '''The number of divisors of n.
+    '''
+    return len(divisors(n))
+
+
+# ------------------------- TEST -------------------------
+# main :: IO ()
+def main():
+    '''The first 100 terms of OEIS A000005.
+       (Shown in rows of 10)
+    '''
+    terms = take(100)(
+        oeisA000005()
+    )
+    columnWidth = 1 + len(str(max(terms)))
+
+    print(
+        '\n'.join(
+            ''.join(
+                str(term).rjust(columnWidth)
+                for term in row
+            )
+            for row in chunksOf(10)(terms)
+        )
+    )
+
+
+# ----------------------- GENERIC ------------------------
+
+# chunksOf :: Int -> [a] -> [[a]]
+def chunksOf(n):
+    '''A series of lists of length n, subdividing the
+       contents of xs. Where the length of xs is not evenly
+       divible, the final list will be shorter than n.
+    '''
+    def go(xs):
+        return [
+            xs[i:n + i] for i in range(0, len(xs), n)
+        ] if 0 < n else None
+    return go
+
+
+# divisors :: Int -> [Int]
+def divisors(n):
+    '''List of all divisors of n including n itself.
+    '''
+    root = floor(sqrt(n))
+    lows = [x for x in range(1, 1 + root) if 0 == n % x]
+    return lows + [n // x for x in reversed(lows)][
+        (1 if n == (root * root) else 0):
+    ]
+
+
+# take :: Int -> [a] -> [a]
+# take :: Int -> String -> String
+def take(n):
+    '''The prefix of xs of length n,
+       or xs itself if n > length xs.
+    '''
+    def go(xs):
+        return (
+            xs[0:n]
+            if isinstance(xs, (list, tuple))
+            else list(islice(xs, n))
+        )
+    return go
+
+
+# MAIN ---
+if __name__ == '__main__':
+    main()