def power_partitions(n,k):
"""
Partitions of n into powers of k in reverse lexicographic order\n
Finite generator
"""
if n < 0:
raise ValueError("n must be non-negative")
if k < 1:
raise ValueError("k must be positive")
if n == k:
yield (n,)
if n == 0:
yield ()
if n == 1:
yield (1,)
else:
for p in powers(k):
if p >= n:
break
while p >= k:
p = p//k
for S in power_partitions(n-p,k):
if S[0] <= p:
yield (p,) + S
def composition_count():
"""
Number of compositions (ordered partition) for each natural (2**(n-1) for n > 0)\n
OEIS A011782
"""
yield 1
for p in powers(2):
yield p
def binary_length():
"""
Binary Length: Bits in the binary representation of each non-negative integer\n
OEIS A070939
"""
yield 1
yield 1
for n, p in enumerate(offset(powers(2), 1), 2):
for i in range(p):
yield n
def ruler():
"""
Ruler Sequence: Largest power of 2 dividing each positive integer, also height of gradation at each position of an infinite ruler\n
OEIS A000120
"""
for e in offset(evens(), 1):
yield 0
for n, p in enumerate(powers(2)):
if e % p != 0:
yield n - 1
break
def base_length(B=10):
"""
Binary Length: Symbols in the representation of each non-negative integer in base B\n
OEIS A070939
"""
require_integers(["B"], [B])
require_geq(["B"], [B], 2)
for i in range(B):
yield 1
for n, p in enumerate(offset(powers(B), 1), 2):
for i in range(p):
yield n
def bin_log_digits(n):
"""
Decimal digits of the base 2 logarithm of n
"""
num = n
den = 1
while True:
int_part = num // den
for e, p in enumerate(powers(2), -1):
if p > int_part:
break
yield e
den = den * 2**e
num = num**10
den = den**10
g = gcd(num, den)
num, den = num // g, den // g