def genNdigitNPower( limit=30 ):
"""
>>> s5= list( genNdigitNPower(6) )
>>> (7, 5, 16807) in s5
True
>>> s9= list( genNdigitNPower(10) )
>>> (8, 9, 134217728) in s9
True
"""
for p in range(limit):
i= 0
while len(digits(i**p)) <= p:
if len(digits(i**p)) == p:
yield i, p, i**p
i += 1
def primeGapPerm():
"""There are just two answers.
>>> from euler49 import primeGapPerm
>>> list(primeGapPerm())
[(1487, 4817, 8147), (2969, 6299, 9629)]
"""
p4 = list(makePrimes())
gaps= set( makeGaps(p4) )
step3= list( threeStep( p4, gaps ) )
for a, b, c in step3:
da= digits(a)
db= digits(b)
dc= digits(c)
if isPermutation(da,db) and isPermutation(db,dc):
yield a,b,c
def genUnorthodoxCancels():
"""Compare digit cancellation and proper fraction reduction.
>>> from euler33 import genUnorthodoxCancels
>>> list( genUnorthodoxCancels() )
[(16, 64), (26, 65), (19, 95), (49, 98)]
"""
for d in range(11,100):
for n in range(11,d):
dn, dd = digits(n), digits(d)
if dn[1] == 0 or dd[1] == 0: continue
n2, d2 = cancelDigits(dn,dd)
if dn == n2: continue # No improper digit cancellation
r= gcd(n,d)
r2= gcd(n2[0],d2[0])
if n//r == n2[0]//r2 and d//r == d2[0]//r2:
#print( n, d, n2, d2 )
yield n, d
def pow2_digits3( n ):
"""Return digits from :math:`2^n`
>>> from euler16 import pow2_digits3
>>> pow2_digits3(15)
[3, 2, 7, 6, 8]
>>> sum(pow2_digits3(15))
26
"""
return digits(2**n)
def numerator( convergent, position ):
"""
>>> converge_e = fraction_steps( 2, tuple(den_pat_e(10)) )
>>> n= numerator( converge_e, 10 )
>>> n
[1, 4, 5, 7]
>>> sum( n )
17
"""
[ next(convergent) for i in range(position-1) ]
term= next(convergent)
#print( term, float(term) )
numerator = term.numerator
return digits(numerator)
def digit_sum_gen():
"""
>>> from euler56 import digit_sum_gen
>>> max( digit_sum_gen() )
(972, 99, 95)
>>> sum(digits(99**95))
972
"""
for a in range(100):
p = 1
for b in range(100):
s = sum(digits(p))
yield s, a, b
p *= a
def prodConcat( n ):
"""Create a 9-digit concatenaed product by successive multiplications
of n by 1, 2, 3, ..., 9
>>> from euler38 import prodConcat
>>> prodConcat( 192 )
[1, 9, 2, 3, 8, 4, 5, 7, 6]
>>> prodConcat( 9 )
[9, 1, 8, 2, 7, 3, 6, 4, 5]
"""
seq= []
for p in range(1,10):
seq.extend( digits(n*p) )
if len(seq) >= 9: break
return seq
def reverseAndAdd( n ):
"""
>>> from euler55 import reverseAndAdd
>>> from euler04 import palindrome
>>> reverseAndAdd(47)
121
>>> palindrome( reverseAndAdd(47) )
True
>>> reverseAndAdd( reverseAndAdd( reverseAndAdd( 349 ) ) )
7337
>>> palindrome( reverseAndAdd( reverseAndAdd( reverseAndAdd( 349 ) ) ) )
True
"""
d = digits(n)
d.reverse()
return n + number(d)
def digitStream( limit ):
"""
>>> from euler40 import digitStream
>>> s= list(digitStream(32))
>>> s[11]
1
>>> s
[1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1]
"""
n= 1
dFed= 0
while dFed < limit:
for d in digits(n):
yield d
dFed += 1
n += 1
def circularPrimes( limit ):
"""
>>> from euler35 import circularPrimes
>>> cp100= circularPrimes( 100 )
>>> len(cp100)
13
>>> sorted( set( cp100 ) )
[2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97]
"""
cp= set()
for n in range(2,limit):
if isprime(n) and n not in cp:
rs= [ number(r) for r in rotations( digits(n) ) ]
circular= all( map( isprime, rs ) )
if circular:
for r in rs:
cp.add(r)
return cp
def answer():
for i in range(1,1000000000):
di= digits(i)
p = [ isPermutation( di, digits(k*i) ) for k in range(2,7) ]
if all(p):
return i
def curious( limit=999999 ):
for i in range(3,limit+1):
p = sum( map( fact, digits(i) ) )
#print i, p
if p == i:
yield i
def answer():
for a,b,c in primeGapPerm():
if a == 1487 and b == 4817 and c == 8147:
continue
return number( digits(a) + digits(b) + digits(c) )
def test():
import doctest
doctest.testmod(verbose=0)
assert pandigital( digits(192384576) )