def middle_square_weyl(n,k,m):
"""
Middle Square Method Augmented with a Weyl Sequence
Args:
n -- seed value
k -- initial value for Weyl Sequence
m -- modulus of Weyl Sequence
"""
# Only n has to be checked since the weyl() function will check k and m
require_integers(["n"],[n])
W = weyl(k,m)
N = int_to_digits(n)
width = len(N)+(len(N)%2)
for w in W:
a = n*n+w
A = int_to_digits(a)
if len(A) % 2 == 1:
A = [0] + A
n = digits_to_int(A[width//2:-width//2])
yield n
def root_digits(n, a, B=10):
"""
Digits of the nth root of a in base B\n
OEIS
"""
require_integers(["a", "n", "B"], [a, n, B])
require_geq(["a"], [a], 0)
require_geq(["n", "B"], [n, B], 2)
Bpow = B**n
chunks = int_to_digits(a, Bpow)
r, y = 0, 0
for d in chain(chunks, constant(0)):
c = Bpow * r + d
x = 0
for i in range(B):
x += 1
if (B * y + x)**n - (Bpow * y**n) > c:
x -= 1
break
y1 = B * y + x
r1 = c - ((B * y + x)**n - (Bpow * y**n))
r, y = r1, y1
yield x
def sqrt_digits(n, B=10):
"""
Digits of the square root of n in base B\n
OEIS A002193, A002194, A002163, A010465, A010466, A010467, A010464,
A010472, A010468, A010490, A010469, A010470, A010471, A010477,
A010473, A010524, A010474
"""
require_integers(["n", "B"], [n, B])
require_geq(["n"], [n], 0)
require_geq(["B"], [B], 2)
Bsq = B * B
p, r = 0, 0
chunks = int_to_digits(n, Bsq)
for d in chain(chunks, constant(0)):
c = Bsq * r + d
x = -1
for i in range(B):
x += 1
if x * (x + (2 * B * p)) > c:
x -= 1
break
r = c - x * (x + 2 * B * p)
p = B * p + x
yield x
def stanley():
"""
Stanley Numbers: Naturals that contain no 2s in their ternary expansion\n
OEIS A005836
"""
for n in naturals():
D = int_to_digits(n, 3)
if 2 not in D:
yield n
def cantor():
"""
Cantor Numbers: Naturals that contain no 1s in their ternary expansion\n
OEIS A005823
"""
for n in naturals():
D = int_to_digits(n, 3)
if 1 not in D:
yield n
def champernowne_digits(B=10):
"""
Digits of the base B version of Champernowne's constant\n
OEIS A003137, A030302, A030548, A033307, A030373, A031219, A030998,
A031035, A031076
"""
require_integers(["B"], [B])
require_geq(["B"], [B], 2)
for n in naturals(1):
yield from iter(int_to_digits(n, B))
def digit_including_words(d, B=10):
"""
Naturals that do contain the digit d in base B, returns tuples\n
OEIS
"""
if d >= B:
raise Exception("d must be less than B")
for n in naturals():
D = int_to_digits(n, B)
if d in D:
yield D
def digit_avoiding_words(d, B=10):
"""
Naturals that do not contain the digit d in base B, returns tuples\n
OEIS A005823, A005836
"""
if d >= B:
raise Exception("d must be less than B")
for n in naturals():
D = int_to_digits(n, B)
if d not in D:
yield D
def radix_digits(B=10):
"""
The digits of each natural number in base B\n
OEIS
"""
require_integers(["B"], [B])
require_geq(["B"], [B], 2)
yield (0, )
for n in naturals(1):
yield int_to_digits(n, B)
def middle_square(n):
"""
Middle Square Method
Args:
n -- seed value
"""
require_integers(["n"],[n])
N = int_to_digits(n)
width = len(N)+(len(N)%2)
while True:
a = n*n
A = int_to_digits(a)
if len(A) % 2 == 1:
A = [0] + A
n = digits_to_int(A[width//2:-width//2])
yield n
def metallic_ratio_digits(n, B=10):
"""
Digits of the Nth Metallic Ratio in base B\n
OEIS A001622, A014176, A098316, A098317, A098318, A176398, A176439,
A176458, A176522
"""
# Prepend zeroes to make addition line up
dign = digits(n, B)
digs = digits(isqrt(n * n + 4), B)
if dign > digs:
N = [0] * (dign - digs) + int_to_digits(n, B)
elif dign < digs:
N = [0] * (digs - dign) + int_to_digits(n, B)
else:
N = int_to_digits(n, B)
# (n + √(n+4))/2
sq = sqrt_digits(n * n + 4, B)
sm = real_sum(N, sq, B)
M = real_div_nat(sm, 2, B)
yield from dropwhile(lambda x: x == 0, M)
def sequence_including_words(S, B=10):
"""
Naturals that do include the sequence S as a subsequence of digits in base B, returns tuples
"""
for s in S:
if s >= B:
raise Exception("all values in S must be less than D")
l = len(S)
for n in naturals():
D = int_to_digits(n, B)
d = len(D)
if d < l:
continue
elif _subseq(S, D):
yield D
def sequence_avoiding(S, B=10):
"""
Naturals that do not include the sequence S as a subsequence of digits in base B
"""
for s in S:
if s >= B:
raise Exception("all values in S must be less than D")
l = len(S)
for n in naturals():
D = int_to_digits(n, B)
d = len(D)
if d < l:
yield n
elif not _subseq(S, D):
yield n
def quadratic_irrational(a, b, c, d, B=10):
"""
Digits of (a+b√c)/d
"""
require_integers(["a", "b", "c", "d", "B"], [a, b, c, d, B])
require_geq(["b", "c", "d"], [b, c, d], 1)
require_geq(["B"], [B], 2)
for i in naturals(2):
if c % (i * i) == 0:
raise Exception("c must be squarefree")
if i * i > c:
break
diga = int_to_digits(a, B)
sq = sqrt_digits(c, B)
pr = real_prod_nat(sq, b, B)
sm = real_sum(pr, diga, B)
M = real_div_nat(sm, d, B)
yield from dropwhile(lambda x: x == 0, M)
