def cyclic(l): if digits(l[0])[:2] != digits(l[-1])[2:]: return False for i in range(1,len(l)): if digits(l[i])[:2] != digits(l[i-1])[2:]: return False return True
def seq(k):
for m in range(1, k + 4):
for p in palins(range(10), m):
if p[0] == 0:
continue
n = num(p)
for d in primes.divisors(n):
if len(digits(d)) == k and len(digits(n // d)) == k:
yield n
def problem30(n):
"""
>>> problem30(200000)
443839
"""
return sum([i for i in range(100, n)\
if i == sum(map(lib.call_power, map(lambda x: (x, 5), lib.digits(i))))])
def increasing(n): s = digits(n) for i in range(len(s) - 1): if int(s[i]) > int(s[i+1]): return False return True
def problem20(n):
"""
>>> problem20(100)
648
"""
value = lib.factorial(n)
return sum(lib.digits(value))
def problem16(base, pow):
"""
>>> problem16(2, 1000)
1366
"""
value = lib.power(base, pow)
return sum(lib.digits(value))
def good(p):
if p < 10:
return False
d = digits(p)
for i in range(1, len(d)):
if not primes.isPrime(num(d[:i])):
return False
if not primes.isPrime(num(d[i:])):
return False
return True
def good(p):
if p < 10:
return False
d = digits(p)
for i in range(1, len(d)):
if not primes.isPrime(num(d[:i])):
return False
if not primes.isPrime(num(d[i:])):
return False
return True
def cycle_length(n):
seen = {n:0}
i = 0
while True:
n = sum(map(fact,map(int,digits(n))))
i += 1
if n in seen:
return i
else:
seen[n] = i
def results():
for k in squaresByLen:
squares = squaresByLen[k]
for words in anagramsByLen[k]:
for word1, word2 in combinations(words, 2):
sw = list(swaps(word1, word2))
for sq in squares:
n = num(apply(digits(sq), sw))
if n in squares:
yield max(n, sq)
def f89(n): if n == 89: return True elif n <= 1: return False else: d = digits(n) d = list(map(int,d)) d = list(map(sq,d)) d = sum(d) return f89(d)
def seq(): yield 81 mem = [] push = lambda b, e: heappush(mem, (b ** e, b, e)) push(2, 3) while True: n, b, e = heappop(mem) if b == sum(digits(n)): yield n if e == 3: push(b + 1, 3) push(b, e + 1)
def main(limit): sq = [d * d for d in range(10)] mem = [None] * limit mem[0] = False mem[1] = False mem[89] = True for n in range(limit): m = n while mem[m] == None: m = sum(sq[d] for d in digits(m)) mem[n] = mem[m] return sum(mem)
def main(limit):
sq = [d * d for d in range(10)]
mem = [None] * limit
mem[0] = False
mem[1] = False
mem[89] = True
for n in range(limit):
m = n
while mem[m] == None:
m = sum(sq[d] for d in digits(m))
mem[n] = mem[m]
return sum(mem)
def seq():
yield 81
mem = []
push = lambda b, e: heappush(mem, (b**e, b, e))
push(2, 3)
while True:
n, b, e = heappop(mem)
if b == sum(digits(n)):
yield n
if e == 3:
push(b + 1, 3)
push(b, e + 1)
def main(indices): index = 0 pos = 1 rv = 1 for n in count(1): d = digits(n) while indices[index] - pos < len(d): rv *= d[indices[index] - pos] index += 1 if index == len(indices): return rv pos += len(d) return rv
def main(indices):
index = 0
pos = 1
rv = 1
for n in count(1):
d = digits(n)
while indices[index] - pos < len(d):
rv *= d[indices[index] - pos]
index += 1
if index == len(indices):
return rv
pos += len(d)
return rv
def main(file):
# trick: for every anagram (word1, word2) calculate the character swaps
# required to turn word2 into word1 then perform those swaps on a square
# number of the same length as word1 and see if the resulting number is
# a square. Only caveat is that the eligible square numbers must have
# at most one of each digit
with open(file, 'r') as f:
words = f.read().replace('"', '').split(',')
# group the words into lists of anagrams
anagrams = dict()
for word in words:
anagrams.setdefault(''.join(sorted(word)), []).append(word)
# group the anagram lists by the length of the word
anagramsByLen = dict()
for k, v in anagrams.items():
if len(v) > 1:
anagramsByLen.setdefault(len(k), []).append(v)
# create eligible squares and group them by length
squaresByLen = dict()
for n in count(4):
sq = n * n
key = numLen(sq)
if key not in anagramsByLen:
break
if key == len(set(digits(sq))): # no repeated digits
squaresByLen.setdefault(key, []).append(sq)
def results():
for k in squaresByLen:
squares = squaresByLen[k]
for words in anagramsByLen[k]:
for word1, word2 in combinations(words, 2):
sw = list(swaps(word1, word2))
for sq in squares:
n = num(apply(digits(sq), sw))
if n in squares:
yield max(n, sq)
return max(results())
#!/usr/bin/env python3 from lib import digits equal = False x = 1 while not equal: equal = True dig = sorted(digits(x)) for mul in range(2,7): if dig != sorted(digits(mul*x)): equal = False x = x + 1 print(x-1)
def main(lim): return max(sum(digits(pow(a, b))) for a in range(lim) for b in range(lim))
def main(lim):
return max(sum(digits(pow(a, b))) for a in range(lim) for b in range(lim))
def seq(n, lim=54):
for _ in range(lim):
n += num(reversed(digits(n)))
yield n
#!/usr/bin/env python3
from lib import digits
cubes = {}
i = 1
while True:
if "".join(sorted(digits(i**3))) not in cubes:
cubes["".join(sorted(digits(i**3)))] = []
cubes["".join(sorted(digits(i**3)))].append(i)
if len(cubes["".join(sorted(digits(i**3)))]) == 5:
print(cubes["".join(sorted(digits(i**3)))])
break
i += 1
##print(cubes)
def dsum(n):
n = pow(Decimal(n), Decimal('0.5'))
n *= pow(10, 100 - numLen(int(n)))
n = int(n)
return sum(digits(n))
def bouncy(n): d = digits(n) return not increasing(d) and not decreasing(d)
def pd(n, k): # nth digit in concatenation of all padded k-digit numbers
q, r = divmod(n, k)
return digits(q)[r]
def dsum(n):
n = pow(Decimal(n), Decimal('0.5'))
n *= pow(10, 100 - numLen(int(n)))
n = int(n)
return sum(digits(n))
def main(n):
return sum(digits(pow(2, n)))
def pd(n, k): # nth digit in concatenation of all padded k-digit numbers q, r = divmod(n, k) return digits(q)[r]
#!/usr/bin/env python3 from lib import digits count= 0 for exp in range(1,25): for digit in range(1,10): if len(digits(digit**exp)) == exp: print(digit, exp, digit**exp) count += 1 print(count)
def main(lim): n, _ = gen(lim) return sum(digits(n))
def seq(n, lim = 54): for _ in range(lim): n += num(reversed(digits(n))) yield n
def main(n, k): d = digits(n) return max(multiply(d[i: i + k]) for i in range(len(d) - k))
def main(n): return sum(digits(factorial(n)))
def main(nums, k):
return num(digits(sum(nums))[:k])
def main(n): return sum(digits(pow(2, n)))
def main(lim):
n, _ = gen(lim)
return sum(digits(n))
def main(nums, k): return num(digits(sum(nums))[:k])
def circular(p):
d = digits(p)
return all(
primes.isPrime(num(d[i:] + d[:i])) for i in range(1, len(d)))
def circular(p): d = digits(p) return all(primes.isPrime(num(d[i:] + d[:i])) for i in range(1, len(d)))
def bouncy(n):
d = digits(n)
return not increasing(d) and not decreasing(d)
def main(n):
return sum(digits(factorial(n)))
if n == 89: return True elif n <= 1: return False else: d = digits(n) d = list(map(int,d)) d = list(map(sq,d)) d = sum(d) return f89(d) e89 = [-1] * (10**8+1) e89[89] = 1 e89[1] = 0 count = 0 for i in range(1,10**7): n = i #print(n) while e89[n] < 0: d = digits(n) d = list(map(int,d)) d = list(map(sq,d)) n = sum(d) e89[i] = e89[n] if e89[i] > 0: #print(i) count += 1 print(count)
while p < 10000:
p = i*(3*i-2)
if p >= 1000 and p < 10000:
p8.append(int(p))
i += 1
for i3 in p3:
for i4 in p4:
for i5 in p5:
for i6 in p6:
l = [i3,i4,i5,i6]
ends = set([])
starts = set([])
for i in l:
ends.add("".join(digits(i)[2:]))
starts.add("".join(digits(i)[:2]))
if len(starts & ends) > 0:
for i7 in p7:
l = [i3,i4,i5,i6,i7]
ends = set([])
starts = set([])
for i in l:
ends.add("".join(digits(i)[2:]))
starts.add("".join(digits(i)[:2]))
if len(starts & ends) > 3:
for i8 in p8:
l = [i3,i4,i5,i6,i7,i8]
ends = set([])
starts = set([])
for i in l:
#!/usr/bin/env python3 from lib import digits a = 3 b = 2 count = 0 for it in range(1, 1001): # print(a,b) a,b = 2*b+a,b+a if len(digits(a)) > len(digits(b)): count += 1 print(count)
def nonRepeatLen(n): mem = set() while n not in mem: mem.add(n) n = sum(map(factorial, digits(n))) return len(mem)