def tribonnaci_vectors():
"""
Unique representation of each natural in the tribonacci base\n
OEIS A278038
"""
trib = offset(tribonacci(), 4)
T = [1]
yield (0, )
for n in naturals(1):
if T[-1] < n:
T.append(next(trib))
out = []
for t in reversed(T):
if n >= t:
out.append(1)
n -= t
else:
if 1 in out:
out.append(0)
yield tuple(out)
def odd_primes():
"""
Odd Primes: Primes that are odd positive integer\n
OEIS A065091
"""
yield from offset(primes(), 1)
def tribonnaci_partitions():
"""
Unique representation of each natural as a sum of distrinct tribonacci numbers\n
OEIS
"""
trib = offset(tribonacci(),4)
T = [1]
yield (0,)
for n in naturals(1):
if T[-1] < n:
T.append(next(trib))
out = []
for t in reversed(T):
if n >= t:
out.append(t)
n -= t
if n == 0:
break
yield tuple(out)
def sparsely_totient():
"""
Positive integers which have a totient less than all greater integers\n
OEIS A036913
"""
P = offset(primorial(),1)
ctr = next(P)
N = []
T = []
def find_sparse(N,T):
for i,t in enumerate(T[:-1]):
if t < min(T[i+1:]):
yield N[i]
for n,t in enumerate(totients(),1):
N.append(n)
T.append(t)
if n == ctr:
ctr = next(P)
yield from find_sparse(N,T)
yield n
N = []
T = []
def square_pyramidal():
"""
Square Pyramidal Numbers: Positive integers that take the shape of a square based pyramid\n
OEIS A000330
"""
S = 1
for s in offset(square(), 2):
yield S
S += s
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 alternating_factorials_1():
"""
Alternating Factorial Numbers: Absolute value of each term\n
OEIS A005165
"""
cyc = sign_sequence(1)
F = offset(factorials(), 1)
out = 0
for f, s in zip(F, cyc):
yield abs(out)
out += f * s
def semifibonacci():
"""
The Semi-Fibonacci Sequence\n
OEIS A030067
"""
L = [1]
for e in offset(evens(),1):
yield L[-1]
L.append(L[e//2-1])
yield L[-1]
L.append(L[e-1] + L[e-2])
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 wilson():
"""
Wilson's Sequence: (n-1)! % n, for primes p-1, otherwise 0 except at four where it equals 2\n
OEIS A061006
"""
yield 0
yield 1
yield 2
yield 2
oldp = 5
for p in offset(primes(), 2):
for i in range(p - oldp - 1):
yield 0
yield p - 1
oldp = p
def two_square():
"""
Sums of exactly two non-zero squares\n
OEIS A000404
"""
sequence = offset(square(), 1)
sqrs = []
sums = []
for s in sequence:
sqrs.append(s)
for i in sqrs:
sums.append(i + s)
sums = sorted(sums)
while sums[0] <= s:
yield sums.pop(0)
def totient_count():
"""
Number of positive integers with totient n for positive n\n
OEIS A014197
"""
D = defaultdict(lambda:0)
lo = 1
S = offset(sparsely_totient(),1)
ctr = next(S)
for n,t in enumerate(totients(),1):
D[t] += 1
if n == ctr:
ctr = next(S)
for i in range(lo,t):
yield D[i]
del D[i]
lo = t