def palindrome(B=10):
"""
Palindrome Numbers: Non-negative integers that are palindromes in base B\n
OEIS A002113, A006995
"""
require_integers(["B"], [B])
require_geq(["B"], [B], 2)
if B == 10:
for n in naturals():
if str(n) == str(n)[::-1]:
yield n
else:
for n in naturals():
m = n
d = []
while n != 0:
n, r = divmod(n, B)
d.append(str(r))
s = "".join(d)
if s == s[::-1]:
yield m
def collatz_length():
"""
Collatz Sequence Lengths: Steps until the Collatz function equals 1 for each positive natural\n
OEIS A006577, A008908
"""
D = {1:0}
for n in naturals(1):
if n in D:
yield D[n]
else:
ctr = n
length = 0
seen = []
while ctr not in D:
ctr = _collatz_step(ctr)
length += 1
seen.append(ctr)
length = D[ctr] + length
D[n] = length
yield D[n]
for num in seen:
length -= 1
D[num] = length
def perfect_powers():
"""
Perfect Powers: Non-negative integers that can be written as a perfect power\n
OEIS A001597 (when offset by 1)
"""
yield 0
yield 1
for n in naturals(2):
lim = floor(log2(n)) + 2
for i in primes():
ctr = 2
while True:
pwr = ctr**i
if pwr > n:
break
if pwr == n:
yield n
break
ctr += 1
if i > lim:
break
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 collatz_highwater():
"""
High-Water Mark of the Collatz Sequences: Record values for the high points of Collatz sequences\n
OEIS A006885
"""
D = {1 : 1}
rec = 0
yield 1
for n in naturals(2):
cur_rec = 0
val = n
while val not in D:
cur_rec = max(cur_rec,val)
val = _collatz_step(val)
cur_rec = max(cur_rec,val)
D[n] = max(D[val],cur_rec)
if cur_rec > rec:
yield D[n]
rec = cur_rec
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 partition_count():
"""
Partition Numbers: Number of unique multisets of positive integers with sum n\n
OEIS A000041
"""
D = [1]
for n in naturals(1):
yield D[-1]
P = gen_pentagonal()
next(P)
sign = -1
k=0
for ctr,i in enumerate(P):
if n-i < 0:
D.append(k)
break
if ctr % 2 == 0:
sign *= -1
k += sign*D[n-i]
def totients():
"""
Totients: Count of positive integers coprime to each positive integer\n
OEIS A000010
"""
D = defaultdict(list)
yield 1
for q in naturals(2):
if q not in D:
yield q-1
D[q + q] = [q]
else:
n,d = 1,1
for p in D[q]:
D[p+q].append(p)
n *= (p-1)
d *= p
yield q*n//d
del D[q]
def primitive_pythagorean_triples():
"""
All primitive pythagorean triples with hypotenuse in non-decreasing order and otherwise lexicographic order\n
OEIS
"""
triples = []
for m in naturals(2):
lim = m * m + 1
for n in range(1 + m % 2, m, 2):
if gcd(m, n) == 1:
a = m * m - n * n
b = 2 * m * n
c = m * m + n * n
triples.append((min(a, b), max(a, b), c))
# This sorting assures that the hypotenuse is increasing and that
# the if the hypotenuse is the same for two tuples the one with the
# smaller initital term will go first
triples.sort(key=lambda x: (x[2], x[0]))
for i in triples:
if i[2] > lim:
break
yield triples.pop(0)
def primitive_hypotenuse():
"""
Primitive Hypotenuse Numbers: Positive integers that can be the hypotenuse of a primitive Pythagorean triple\n
OEIS A008846
"""
L = []
for m in naturals(2):
lim = m * m + 1 # smallest number that can be produced in this round
for n in range(1 + m % 2, m, 2):
if gcd(m, n) == 1:
p = m * m + n * n
if p not in L:
L.append(p)
L.sort()
for i in L:
if i > lim:
break
yield L.pop(0)
def roman_numerals_str():
"""
The positive integers as standard Roman Numerals, returns strings
"""
for n in naturals(1):
yield int_to_roman(n)
def leibniz_harmonic_triangle(flatten=False):
"""
Leibniz's Harmonic Triangle: Each term is the sum of the two below\n
Args:
flatten -- boolean, if True returns one number at a time top to bottom left to right, otherwise returns on row at a time
A003506
"""
if flatten:
for row in leibniz_harmonic_triangle(flatten=False):
yield from row
else:
row = [1]
for n in naturals(2):
yield tuple(row)
new_row = [Fraction(1,n)]
for term in row:
new_row.append(term-new_row[-1])
row = new_row
def juggler_highwater():
"""
High-Water Mark of the Juggler Sequences: Record values for the high points of Juggler sequences\n
OEIS A143745
"""
D = {1: 1}
rec = 0
yield 1
for n in naturals(2):
cur_rec = 0
val = n
while val not in D:
cur_rec = max(cur_rec, val)
val = _juggler_step(val)
cur_rec = max(cur_rec, val)
D[n] = max(D[val], cur_rec)
if cur_rec > rec:
yield D[n]
rec = cur_rec
def wedderburn_etherington():
"""
Wedderburn-Etherington Numbers\n
OEIS A001190
"""
S = [0,1,1]
yield from S
for n in naturals(2):
# Odd step
t = 0
for i in range(1,n):
t += S[i]*S[2*n-i-1]
yield t
S.append(t)
# Even step
t = (S[n]*(S[n]+1))//2
for i in range(1,n):
t += S[i]*S[2*n-i]
yield t
S.append(t)
def dyck_language():
"""
All the Dyck words with 'up' as +1 and 'down' as -1
"""
for n in naturals(1):
yield from dyck_words(n)
def dyck_language_str():
"""
All words consisting of correctly matched pairs of parentheses
"""
for n in naturals(1):
yield from dyck_words_str(n)
def juggler_length():
"""
Juggler Sequence Lengths: Steps until the Juggler Sequence starting at each positive natural reaches 1\n
OEIS A007320
"""
D = {1: 0}
for n in naturals(1):
if n in D:
yield D[n]
else:
ctr = n
length = 0
seen = []
while ctr not in D:
ctr = _juggler_step(ctr)
length += 1
seen.append(ctr)
length = D[ctr] + length
D[n] = length
yield D[n]
for num in seen:
length -= 1
D[num] = length
def prime_signatures():
"""
Prime Signatures: The prime signature of each positive integer, returns tuples\n
OEIS A124010
"""
D = defaultdict(list)
yield ()
for q in naturals(2):
if q not in D:
yield (1, )
D[q + q] = [q]
else:
T = []
r = q
for p in D[q]:
T.append(0)
D[p + q].append(p)
while r % p == 0:
r //= p
T[-1] += 1
yield tuple(T)
del D[q]
def reflexive_relation():
"""
Number of Reflexive Relations on n elements\n
OEIS A053763
"""
for n in naturals():
yield 2**(n * n - n)
def sum_free_subset():
"""
The number of sum-free subsets of the set {1...n} for all naturals n
OEIS A007865
"""
for n in naturals():
yield len([i for i in sum_free_subsets(n)])
def central_binomial():
"""
Central Binomial Coefficients: Middle term of the odd rows of Pascal's Triangle\n
OEIS A000984
"""
for n in naturals():
yield comb(2*n,n)
def cake():
"""
Cake Numbers: Maximum number of pieces produced when cutting a sphere with exactly n planes\n
OEIS A000125
"""
for n in naturals():
yield (n*n*n+5*n+6)//6
def lazy_caterer():
"""
Lazy Caterer Numbers: Maximum number of pieces produced when cutting a circle with exactly n lines\n
OEIS A000124
"""
for n in naturals():
yield (n*(n+1))//2+1
def aliquot():
"""
Aliquot Numbers: Sum of proper divisors for each positive integer\n
OEIS A001065
"""
for n in naturals(1):
yield aliquot_sum(n)
def deficiency():
"""
Deficiency: Each positive integer minus its aliquot sum\n
OEIS A033879
"""
for n in naturals(1):
yield n-aliquot_sum(n)
def abundance():
"""
Abundance: The aliquot sum of each positive number minus that number\n
OEIS A033880
"""
for n in naturals(1):
yield aliquot_sum(n)-n
def relation():
"""
Number of Binary Relations on n elements\n
OEIS A002416
"""
for n in naturals():
yield 2**(n * n)
def combinadic(k):
"""
k-Combinadic Numbers: Natural numbers represented as descending combinations of k elements\n
OEIS
"""
for n in naturals():
yield int_to_comb(n,k)
def even_odd():
"""
Positive integers but with odds and evens exchanged\n
OEIS A103889
"""
for n, s in zip(naturals(1), sign_sequence(1)):
yield n + s
def all_quadratic_nonresidues():
"""
Irregular array by rows listing all the quadratic nonresidues for each positive natural\n
OEIS A096008
"""
for n in naturals(1):
yield from quadratic_nonresidue(n)