def taylor_convergents(A,c=0,x=0):
"""
Convergents of the Talyor Series defined by the sequence A and constant c, at x
"""
require_iterable(["A"],[A])
c = Fraction(c)
x = Fraction(x)
for P in taylor_sums(A,c):
yield poly_eval(P,x)
def dirichlet_terms(A,s):
"""
Terms of the given Dirichlet series
"""
require_integers(["s"], [s])
require_geq(["s"], [s], 1)
require_iterable(["A"],[A])
for n,a in enumerate(A,1):
f = Fraction(a)/Fraction(n**s)
yield f
def taylor_terms(A, c=0):
"""
Coefficients of each term of the Talyor Series defined by the sequence A and constant c\n
"""
require_iterable(["A"], [A])
require_integers(["c"], [c])
P = [1]
for a in A:
yield poly_mult([a], P)
P = poly_mult([1, -c], P)
def taylor_sums(A, c=0):
"""
Coefficients of each partial sum of the Talyor Series defined by the sequence A and constant c\n
"""
require_iterable(["A"], [A])
require_integers(["c"], [c])
S = [0]
for P in taylor_terms(A, c):
S = poly_sum(P, S)
yield S
def taylor_sums(A,c=0):
"""
Coefficients of each partial sum of the Talyor Series defined by the sequence A and constant c
"""
require_iterable(["A"],[A])
c = Fraction(c)
S = [Fraction(0)]
for P in taylor_terms(A,c):
S = poly_sum(P,S)
yield S
def taylor_terms(A,c=0):
"""
Coefficients of each term of the Talyor Series defined by the sequence A and constant c
"""
require_iterable(["A"],[A])
P = [Fraction(1)]
Q = [Fraction(1),Fraction(-c)]
for a in A:
yield poly_mult([a],P)
P = poly_mult(Q,P)
def dirichlet_sums(A,s):
"""
Partial Sums of the given Dirichlet series
"""
require_integers(["s"], [s])
require_geq(["s"], [s], 1)
require_iterable(["A"],[A])
p = Fraction(0)
for f in dirichlet_terms(A,s):
p += f
yield p
def prime_product(P, inclusive=False):
"""
Naturals such that all prime factors are in the set P
Args:
P -- an interable consisting of primes
inclusive -- bool, if True only numbers divisible by all elements of P are included
OEIS
"""
require_iterable(["P"], [P])
P = set(P)
I = prod(P)
for p in P:
if not isprime(p):
raise Exception(
f"All elements of P must be prime, {p} is not prime")
if inclusive:
for n in arithmetic(I, I):
m = n
for p in P:
while m % p == 0:
m //= p
if m == 1:
yield n
else:
for n in naturals(2):
m = n
for p in P:
while m % p == 0:
m //= p
if m == 1:
yield n