Trac #16878: faster is_prime

Right now to test if a Sage integer is prime it is faster to call
`prime_range` rather than `.is_prime()`...
{{{
sage: timeit("bool(prime_range(121,122))", number=10000)
10000 loops, best of 3: 1.09 µs per loop
sage:  timeit("bool(prime_range(1009,1010))", number=10000)
10000 loops, best of 3: 1.2 µs per loop
}}}
versus
{{{
sage: timeit("121.is_prime()", number=10000)
10000 loops, best of 3: 5.3 µs per loop
sage: timeit("1009.is_prime()", number=10000)
10000 loops, best of 3: 4.19 µs per loop
}}}

The patch does some tiny modifications in `integer.pyx` and we get
{{{
sage: timeit("121.is_prime()", number=10000)
10000 loops, best of 3: 812 ns per loop
sage: timeit("1009.is_prime()", number=10000)
10000 loops, best of 3: 730 ns per loop
}}}

We also modify `is_prime_power()` to return `False` for 1 and raise an
error for non-integral rationals like `1/2`.

See also [https://groups.google.com/forum/#!topic/sage-devel/v6s2RI4dd1Q
this sage-devel discussion].

URL: http://trac.sagemath.org/16878
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix, Jeroen Demeyer
Reviewer(s): Jeroen Demeyer, Vincent Delecroix

src/sage/combinat/sloane_functions.py

@@ -2390,7 +2390,8 @@ def _eval(self, n):
 class A000015(SloaneSequence):
     def __init__(self):
         r"""
-        Smallest prime power `\geq n`.
+        Smallest prime power `\geq n` (where `1` is considered a prime
+        power).
 
         INPUT:
 
@@ -2447,7 +2448,7 @@ def _eval(self, n):
             sage: [sloane.A000015._eval(n) for n in range(1,11)]
             [1, 2, 3, 4, 5, 7, 7, 8, 9, 11]
         """
-        if arith.is_prime_power(n):
+        if n == 1 or arith.is_prime_power(n):
             return n
         else:
             return arith.next_prime_power(n)

src/sage/misc/functional.py

@@ -463,7 +463,7 @@ def symbolic_sum(expression, *args, **kwds):
             sage: sum(valuation(n,2),n,1,5)
             Traceback (most recent call last):
             ...
-            AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'valuation'
+            TypeError: unable to convert n to an integer
             
         Again, use python ``sum()``::
         

src/sage/rings/all.py

@@ -137,7 +137,7 @@
 
 # Arithmetic
 from arith import algdep, bernoulli, is_prime, is_prime_power, \
-    is_pseudoprime, is_pseudoprime_small_power, valuation, \
+    is_pseudoprime, is_pseudoprime_power, is_pseudoprime_small_power, \
     prime_powers, primes_first_n, eratosthenes, primes, \
     next_prime_power, next_probable_prime, next_prime, \
     previous_prime, previous_prime_power, random_prime, \
@@ -158,7 +158,7 @@
     fundamental_discriminant, squarefree_divisors, \
     Sigma, radical, Euler_Phi, binomial_coefficients, jacobi_symbol, \
     Moebius, dedekind_sum, \
-    prime_factors
+    prime_factors, valuation
 
 
 from fast_arith import prime_range