def amicable_pairs():
"""
Amicable Pairs: Pairs of numbers which are each the Aliquot sum of the other\n
OEIS A063990
"""
for n in naturals(284):
b = aliquot_sum(n)
if b >= n:
continue
if aliquot_sum(b) == n:
yield b,n
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 abundant():
"""
Abundant Numbers: Positive integers that are less than their aliquot sum\n
OEIS A005101
"""
for n in naturals(1):
if aliquot_sum(n) > n:
yield n
def deficient():
"""
Deficient Numbers: Positive integers that are greater than their aliquot sum\n
OEIS A005100
"""
for n in naturals(1):
if aliquot_sum(n) < n:
yield n
def aliquot_recurrence(n):
"""
Aliquot Sequence of n: Recurrence relation of which each term of the aliquot sum of the previous
Args:
n -- integer greater than zero
OEIS
"""
require_integers(["n"],[n])
require_geq(["n"],[n],1)
while True:
yield n
n = aliquot_sum(n)
def untouchable():
"""
Untouchable Numbers: Non-negative integers that cannot be an aliquot sum\n
OEIS A005114
"""
seen = set({})
lo, hi = 3,3
for n in naturals(2):
hi = (n-1)**2
for k in range(lo,hi+1):
seen.add(aliquot_sum(k))
if n not in seen:
yield n
lo = hi
def touchable():
"""
Touchable Numbers: Non-negative integers that can be an aliquot sum\n
OEIS A078923
"""
seen = set({})
lo, hi = 3,3
yield 0
yield 1
for n in naturals(3):
hi = (n-1)**2
for k in range(lo,hi+1):
seen.add(aliquot_sum(k))
if n in seen:
yield n
lo = hi