def practical():
"""
Practical Numbers: Positive integers with factors that can be summed to any smaller positive integer\n
OEIS A005153
"""
def srinivasan_ineq(a):
F = prime_factorization(a)
C = Counter(F)
P = 2**(C[2]+1)-1
del C[2]
for p,n in C.items():
if p > P+1:
return False
P *= (p**(n+1)-1)//(p-1)
return True
yield 1
for a in arithmetic(2,2):
if srinivasan_ineq(a):
yield a
def selfridge():
"""
Selfridge's Sequence: Used for parameter selection in Lucas pseudoprime tests\n
OEIS
"""
for s, n in zip(sign_sequence(1), arithmetic(5, 2)):
yield s * n
def nonadditive():
"""
No term is the sum of any previous distinct pair\n
OEIS A033627
"""
yield 1
yield 2
for a in arithmetic(4, 3):
yield a
def square():
"""
Square Numbers\n
OEIS A000290
"""
S = 0
for a in arithmetic(1, 2):
yield S
S += a
def oblong():
"""
Oblong Numbers (sometimes Pronic Numbers): Sums of consecutive non-negative integers\n
OEIS A002378
"""
S = 0
for a in arithmetic(2, 2):
yield S
S += a
def e_cfrac():
"""
Terms of the simple continued fraction of Euler's Number
OEIS A003417
"""
M = arithmetic(2, 2)
yield 2
while True:
yield 1
yield next(M)
yield 1
def rectangular(d):
"""
Rectangular Numbers: Generalization of Oblong Numbers\n
OEIS A005563, A028552, A028347, A028557, A028560, A028563, A028566,
A028569, A098603, A119412, A132759, A098847-A098850, A120071,
A132760-A132773
"""
S = 0
require_integers(["d"], [d])
require_geq(["d"], [d], 1)
for a in arithmetic(d + 1, 2):
yield S
S += a
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
def lucky():
"""
Lucky Numbers: Prime-like integers resulting from a modified sieve of Eratosthenes\n
OEIS A000959
"""
yield 1
yield 3
# Terms of the sequence and where in the cycle each is
terms = [3]
ctrs = [2]
# Update the cycles positions
# If any cycle has reached zero we're not at a lucky number and no later
# cycles will be updated since the value has been sieved out before that
# number is known
def update():
for n, t in enumerate(terms):
ctrs[n] = (ctrs[n] + 1) % t
if ctrs[n] == 0:
return False
return True
# Which lucky number we're currently looking for
# This sets where the cycle for a newly found lucky number begins
nth = 3
# Go through the odds greater than 5 looking for lucky numbers
for o in arithmetic(5, 2):
if update():
yield o
terms.append(o)
ctrs.append(nth)
nth += 1