def getNSphereRadius( n, k ):
if real_int( k ) < 3:
raise ValueError( 'the number of dimensions must be at least 3' )
if not isinstance( n, RPNMeasurement ):
return RPNMeasurement( n, 'meter' )
dimensions = n.getDimensions( )
if dimensions == { 'length' : 1 }:
return n
elif dimensions == { 'length' : int( k - 1 ) }:
m2 = n.convertValue( RPNMeasurement( 1, [ { 'meter' : int( k - 1 ) } ] ) )
result = root( fdiv( fmul( m2, gamma( fdiv( k, 2 ) ) ),
fmul( 2, power( pi, fdiv( k, 2 ) ) ) ), fsub( k, 1 ) )
return RPNMeasurement( result, [ { 'meter' : 1 } ] )
elif dimensions == { 'length' : int( k ) }:
m3 = n.convertValue( RPNMeasurement( 1, [ { 'meter' : int( k ) } ] ) )
result = root( fmul( fdiv( gamma( fadd( fdiv( k, 2 ), 1 ) ),
power( pi, fdiv( k, 2 ) ) ),
m3 ), k )
return RPNMeasurement( result, [ { 'meter' : 1 } ] )
else:
raise ValueError( 'incompatible measurement type for computing the radius: ' +
str( dimensions ) )
def getKSphereRadius(n, k):
if k < 3:
raise ValueError('the number of dimensions must be at least 3')
if not isinstance(n, RPNMeasurement):
return RPNMeasurement(n, 'meter')
dimensions = n.getDimensions()
if dimensions == {'length': 1}:
return n
if dimensions == {'length': int(k - 1)}:
area = n.convertValue(RPNMeasurement(1, [{'meter': int(k - 1)}]))
result = root(
fdiv(fmul(area, gamma(fdiv(k, 2))), fmul(2, power(pi, fdiv(k,
2)))),
fsub(k, 1))
return RPNMeasurement(result, [{'meter': 1}])
if dimensions == {'length': int(k)}:
volume = n.convertValue(RPNMeasurement(1, [{'meter': int(k)}]))
result = root(
fmul(fdiv(gamma(fadd(fdiv(k, 2), 1)), power(pi, fdiv(k, 2))),
volume), k)
return RPNMeasurement(result, [{'meter': 1}])
raise ValueError(
'incompatible measurement type for computing the radius: ' +
str(dimensions))
def getSuperRootsOperator(n, k):
'''Returns all the super-roots of n, not just the nice, positive, real one.'''
k = fsub(k, 1)
factors = [fmul(i, root(k, k)) for i in unitroots(int(k))]
base = root(fdiv(log(n), lambertw(fmul(k, log(n)))), k)
return [fmul(i, base) for i in factors]
def getRoot( n, k ):
if isinstance( n, RPNMeasurement ):
if not isInteger( k ):
raise ValueError( 'cannot take a fractional root of a measurement' )
newUnits = RPNUnits( n.getUnits( ) )
for unit, exponent in newUnits.items( ):
if fmod( exponent, k ) != 0:
if k == 2:
name = 'square'
elif k == 3:
name = 'cube'
else:
name = getOrdinalName( k )
raise ValueError( 'cannot take the ' + name + ' root of this measurement: ', n.getUnits( ) ) #print measurement RPNMeasurement.getValue( ) )
newUnits[ unit ] /= k
value = root( n.getValue( ), k )
return RPNMeasurement( value, newUnits )
return root( n, k )
def getRoot( self, operand ):
if ( floor( operand ) != operand ):
raise ValueError( 'cannot take a fractional root of a measurement' )
newUnits = RPNUnits( self.units )
for unit, exponent in newUnits.items( ):
if fmod( exponent, operand ) != 0:
if operand == 2:
name = 'square'
elif operand == 3:
name = 'cube'
else:
name = getOrdinalName( operand )
baseUnits = self.convertToBaseUnits( )
if ( baseUnits != self ):
return baseUnits.getRoot( operand )
else:
raise ValueError( 'cannot take the ' + name + ' root of this measurement: ', self.units )
newUnits[ unit ] /= operand
value = root( self.value, operand )
return RPNMeasurement( value, newUnits ).normalizeUnits( )
def getNthPadovanNumber( arg ):
n = fadd( real( arg ), 4 )
a = root( fsub( fdiv( 27, 2 ), fdiv( fmul( 3, sqrt( 69 ) ), 2 ) ), 3 )
b = root( fdiv( fadd( 9, sqrt( 69 ) ), 2 ), 3 )
c = fadd( 1, fmul( mpc( 0, 1 ), sqrt( 3 ) ) )
d = fsub( 1, fmul( mpc( 0, 1 ), sqrt( 3 ) ) )
e = power( 3, fdiv( 2, 3 ) )
r = fadd( fdiv( a, 3 ), fdiv( b, e ) )
s = fsub( fmul( fdiv( d, -6 ), a ), fdiv( fmul( c, b ), fmul( 2, e ) ) )
t = fsub( fmul( fdiv( c, -6 ), a ), fdiv( fmul( d, b ), fmul( 2, e ) ) )
return nint( re( fsum( [ fdiv( power( r, n ), fadd( fmul( 2, r ), 3 ) ),
fdiv( power( s, n ), fadd( fmul( 2, s ), 3 ) ),
fdiv( power( t, n ), fadd( fmul( 2, t ), 3 ) ) ] ) ) )
def __init__(self, polyA=None, polyB=None):
# Input as list
self.polyA = list(polyA or [0])[:]
self.polyB = list(polyB or [0])[:]
# Remove leading zero coefficients
while self.polyA[-1] == 0:
self.polyA.pop()
self.len_A = len(self.polyA)
while self.polyB[-1] == 0:
self.polyB.pop()
self.len_B = len(self.polyB)
# Add 0 to make lengths equal a power of 2
self.C_max_length = int(2**np.ceil(
np.log2(len(self.polyA) + len(self.polyB) - 1)))
while len(self.polyA) < self.C_max_length:
self.polyA.append(0)
while len(self.polyB) < self.C_max_length:
self.polyB.append(0)
# A complex root used for the fourier transform
self.root = complex(mpmath.root(x=1, n=self.C_max_length, k=1))
# The product
self.product = self.__multiply()
def getRoot(self, operand):
if floor(operand) != operand:
raise ValueError('cannot take a fractional root of a measurement')
newUnits = RPNUnits(self.units)
for unit, exponent in newUnits.items():
if fmod(exponent, operand) != 0:
if operand == 2:
name = 'square'
elif operand == 3:
name = 'cube'
else:
# getOrdinalName( operand )
name = str(int(operand)) + 'th'
baseUnits = self.convertToPrimitiveUnits()
if baseUnits != self:
return baseUnits.getRoot(operand)
raise ValueError(
'cannot take the ' + name + ' root of this measurement: ',
self.units)
newUnits[unit] /= operand
value = root(self.value, operand)
return RPNMeasurement(value, newUnits).normalizeUnits()
def getRoot(n, k):
if isinstance(n, RPNMeasurement):
return n.getRoot(k)
if not isint(k):
return power(n, fdiv(1, k))
return root(n, k)
def getRoot( n, k ):
if isinstance( n, RPNMeasurement ):
return n.getRoot( k )
if not isint( k ):
return power( n, fdiv( 1, k ) )
else:
return root( n, k )
def __call__(self, y):
"""
This function should be used by the root finder.
If the brentq root finder is used then the derivative does not help.
"""
r = self.r2 / self.r1
lhs = self.a1 - self.b1 * mp.root(y, 2)
rhs = self.a2 * mp.power(y, r - 1) + self.b2 * mp.power(y, 1.5*r - 1)
return rhs - lhs
def __call__(self, y):
"""
This function should be used by the root finder.
If the brentq root finder is used then the derivative does not help.
"""
r = self.r2 / self.r1
lhs = self.a1 - self.b1 * mp.root(y, 2)
rhs = self.a2 * mp.power(y, r - 1) + self.b2 * mp.power(y, 1.5 * r - 1)
return rhs - lhs
def calculateGeometricMean( args ):
if isinstance( args, RPNGenerator ):
return calculateGeometricMean( list( args ) )
elif isinstance( args, list ):
if isinstance( args[ 0 ], ( list, RPNGenerator ) ):
return [ calculateGeometricMean( list( arg ) ) for arg in args ]
else:
return root( fprod( args ), len( args ) )
else:
return args
def main():
mpmath.mp.dps = 10000
T = int(sys.stdin.readline())
for t in range(1, T+1):
N, M = map(int, sys.stdin.readline().strip().split())
a = map(mpmath.mpf, sys.stdin.readline().strip().split())
print "Case #%d:" % t
print >> sys.stderr, "Case #%d:" % t
for _ in range(M):
L, R = map(int, sys.stdin.readline().strip().split())
prod = reduce(lambda x, y: x * y, a[L:R+1], 1)
# print >> sys.stderr, "prod", prod, "L", L, "R", R
print float(mpmath.root(prod, R-L+1))
def solveQuarticPolynomialOperator( _a, _b, _c, _d, _e ):
# pylint: disable=invalid-name
'''
This function applies the quartic formula to solve a polynomial
with coefficients of a, b, c, d, and e.
'''
if mp.dps < 50:
mp.dps = 50
# maybe it's really an order-3 polynomial
if _a == 0:
return solveCubicPolynomial( _b, _c, _d, _e )
# degenerate case, just return the two real and two imaginary 4th roots of the
# constant term divided by the 4th root of a
if _b == 0 and _c == 0 and _d == 0:
e = fdiv( _e, _a )
f = root( _a, 4 )
x1 = fdiv( root( fneg( e ), 4 ), f )
x2 = fdiv( fneg( root( fneg( e ), 4 ) ), f )
x3 = fdiv( mpc( 0, root( fneg( e ), 4 ) ), f )
x4 = fdiv( mpc( 0, fneg( root( fneg( e ), 4 ) ) ), f )
return [ x1, x2, x3, x4 ]
# otherwise we have a regular quartic to solve
b = fdiv( _b, _a )
c = fdiv( _c, _a )
d = fdiv( _d, _a )
e = fdiv( _e, _a )
# we turn the equation into a cubic that we can solve
f = fsub( c, fdiv( fmul( 3, power( b, 2 ) ), 8 ) )
g = fsum( [ d, fdiv( power( b, 3 ), 8 ), fneg( fdiv( fmul( b, c ), 2 ) ) ] )
h = fsum( [ e, fneg( fdiv( fmul( 3, power( b, 4 ) ), 256 ) ),
fmul( power( b, 2 ), fdiv( c, 16 ) ), fneg( fdiv( fmul( b, d ), 4 ) ) ] )
roots = solveCubicPolynomial( 1, fdiv( f, 2 ), fdiv( fsub( power( f, 2 ), fmul( 4, h ) ), 16 ),
fneg( fdiv( power( g, 2 ), 64 ) ) )
y1 = roots[ 0 ]
y2 = roots[ 1 ]
y3 = roots[ 2 ]
# pick two non-zero roots, if there are two imaginary roots, use them
if y1 == 0:
root1 = y2
root2 = y3
elif y2 == 0:
root1 = y1
root2 = y3
elif y3 == 0:
root1 = y1
root2 = y2
elif im( y1 ) != 0:
root1 = y1
if im( y2 ) != 0:
root2 = y2
else:
root2 = y3
else:
root1 = y2
root2 = y3
# more variables...
p = sqrt( root1 )
q = sqrt( root2 )
r = fdiv( fneg( g ), fprod( [ 8, p, q ] ) )
s = fneg( fdiv( b, 4 ) )
# put together the 4 roots
x1 = fsum( [ p, q, r, s ] )
x2 = fsum( [ p, fneg( q ), fneg( r ), s ] )
x3 = fsum( [ fneg( p ), q, fneg( r ), s ] )
x4 = fsum( [ fneg( p ), fneg( q ), r, s ] )
return [ chop( x1 ), chop( x2 ), chop( x3 ), chop( x4 ) ]
def solveCubicPolynomial( a, b, c, d ):
# pylint: disable=invalid-name
'''
This function applies the cubic formula to solve a polynomial
with coefficients of a, b, c and d.
'''
if mp.dps < 50:
mp.dps = 50
if a == 0:
return solveQuadraticPolynomial( b, c, d )
f = fdiv( fsub( fdiv( fmul( 3, c ), a ), fdiv( power( b, 2 ), power( a, 2 ) ) ), 3 )
g = fdiv( fadd( fsub( fdiv( fmul( 2, power( b, 3 ) ), power( a, 3 ) ),
fdiv( fprod( [ 9, b, c ] ), power( a, 2 ) ) ),
fdiv( fmul( 27, d ), a ) ), 27 )
h = fadd( fdiv( power( g, 2 ), 4 ), fdiv( power( f, 3 ), 27 ) )
# all three roots are the same
if h == 0:
x1 = fneg( root( fdiv( d, a ), 3 ) )
x2 = x1
x3 = x2
# two imaginary and one real root
elif h > 0:
r = fadd( fneg( fdiv( g, 2 ) ), sqrt( h ) )
if r < 0:
s = fneg( root( fneg( r ), 3 ) )
else:
s = root( r, 3 )
t = fsub( fneg( fdiv( g, 2 ) ), sqrt( h ) )
if t < 0:
u = fneg( root( fneg( t ), 3 ) )
else:
u = root( t, 3 )
x1 = fsub( fadd( s, u ), fdiv( b, fmul( 3, a ) ) )
real = fsub( fdiv( fneg( fadd( s, u ) ), 2 ), fdiv( b, fmul( 3, a ) ) )
imaginary = fdiv( fmul( fsub( s, u ), sqrt( 3 ) ), 2 )
x2 = mpc( real, imaginary )
x3 = mpc( real, fneg( imaginary ) )
# all real roots
else:
j = sqrt( fsub( fdiv( power( g, 2 ), 4 ), h ) )
k = acos( fneg( fdiv( g, fmul( 2, j ) ) ) )
if j < 0:
l = fneg( root( fneg( j ), 3 ) )
else:
l = root( j, 3 )
m = cos( fdiv( k, 3 ) )
n = fmul( sqrt( 3 ), sin( fdiv( k, 3 ) ) )
p = fneg( fdiv( b, fmul( 3, a ) ) )
x1 = fsub( fmul( fmul( 2, l ), cos( fdiv( k, 3 ) ) ), fdiv( b, fmul( 3, a ) ) )
x2 = fadd( fmul( fneg( l ), fadd( m, n ) ), p )
x3 = fadd( fmul( fneg( l ), fsub( m, n ) ), p )
return [ chop( x1 ), chop( x2 ), chop( x3 ) ]
def get_binary(self, item):
def _comma(a, b):
if not isinstance(a, mpmath.matrix):
a = mpmath.matrix([[a]])
if not isinstance(b, mpmath.matrix):
b = mpmath.matrix([[b]])
assert a.rows == b.rows, 'matrix rows not equal'
r = mpmath.matrix(a.rows, a.cols + b.cols)
for i in range(a.rows):
for j in range(a.cols):
r[(i, j)] = a[(i, j)]
for j in range(b.cols):
r[(i, j + a.cols)] = b[(i, j)]
return r
def _semicolon(a, b):
if not isinstance(a, mpmath.matrix):
a = mpmath.matrix([[a]])
if not isinstance(b, mpmath.matrix):
b = mpmath.matrix([[b]])
assert a.cols == b.cols, 'matrix cols not equal'
r = mpmath.matrix(a.rows + b.rows, a.cols)
for j in range(a.cols):
for i in range(a.rows):
r[(i, j)] = a[(i, j)]
for i in range(b.rows):
r[(i + a.rows, j)] = b[(i, j)]
return r
def _mul(a, b):
r = a * b
if isinstance(r, mpmath.matrix) and r.rows == 1 and r.cols == 1:
return r[(0, 0)]
else:
return r
def _dot(a, b):
if isinstance(a, mpmath.matrix) and isinstance(b, mpmath.matrix):
if a.rows == b.rows == 1 and a.cols == b.cols:
return _mul(a, b.transpose())
elif a.cols == b.cols == 1 and a.rows == b.rows:
return _mul(a.transpose(), b)
return _mul(a, b)
def _cross(a, b):
try:
la = OpList.__analyse_triple(a)
lb = OpList.__analyse_triple(b)
r = [
la[1] * lb[2] - la[2] * lb[1],
la[2] * lb[0] - la[0] * lb[2],
la[0] * lb[1] - la[1] * lb[0]
]
if a.cols == b.cols == 1:
return mpmath.matrix([[i] for i in r])
if a.rows == b.rows == 1:
return mpmath.matrix([r])
else:
raise Exception
except:
return _mul(a, b)
return {
'+':
calc2.BinaryOperator('+', lambda x, y: x + y, 30),
'-':
calc2.BinaryOperator('-', lambda x, y: x - y, 30),
'*':
calc2.BinaryOperator('*', _dot, 31),
'x':
calc2.BinaryOperator('x', _cross, 31),
'×':
calc2.BinaryOperator('x', _cross, 31),
'/':
calc2.BinaryOperator('/', lambda x, y: x / y, 31),
'mod':
calc2.BinaryOperator('mod', mpmath.fmod, 31),
'^':
calc2.BinaryOperator('^', mpmath.power, 32),
'rt':
calc2.BinaryOperator('y_/x', lambda y, x: mpmath.root(x, y), 32),
'_/':
calc2.BinaryOperator('y_/x', lambda y, x: mpmath.root(x, y), 32),
'√':
calc2.BinaryOperator('y_/x', lambda y, x: mpmath.root(x, y), 32),
'log':
calc2.BinaryOperator('blogA', lambda b, a: OpList.__log(b, a), 32),
'P':
calc2.BinaryOperator(
'nPr',
lambda n, r: mpmath.factorial(n) / mpmath.factorial(n - r),
50),
'C':
calc2.BinaryOperator(
'nCr', lambda n, r: mpmath.factorial(n) /
(mpmath.factorial(n - r) * mpmath.factorial(r)), 50),
'E':
calc2.BinaryOperator(
'*10^', lambda s, e: mpmath.fmul(s, mpmath.power(10, e)), 70),
',':
calc2.BinaryOperator(',', _comma, 22),
';':
calc2.BinaryOperator(';', _semicolon, 21),
}[item]
def calculateGeometricMean( args ):
if isinstance( args[ 0 ], ( list, RPNGenerator ) ):
return [ calculateGeometricMean( list( arg ) ) for arg in args ]
return root( fprod( args ), len( args ) )
def describeIntegerOperator(n):
indent = ' ' * 4
print()
print(int(n), 'is:')
if isOdd(n):
print(indent + 'odd')
elif isEven(n):
print(indent + 'even')
if isPrime(n):
isPrimeFlag = True
print(indent + 'prime')
elif n > 3:
isPrimeFlag = False
print(indent + 'composite')
else:
isPrimeFlag = False
if isKthPower(n, 2):
print(indent + 'the ' + getShortOrdinalName(sqrt(n)) +
' square number')
if isKthPower(n, 3):
print(indent + 'the ' + getShortOrdinalName(cbrt(n)) + ' cube number')
for i in arange(4, fadd(ceil(log(fabs(n), 2)), 1)):
if isKthPower(n, i):
print(indent + 'the ' + getShortOrdinalName(root(n, i)) + ' ' +
getNumberName(i, True) + ' power')
# triangular
guess = findPolygonalNumber(n, 3)
if getNthPolygonalNumber(guess, 3) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' triangular number')
# pentagonal
guess = findPolygonalNumber(n, 5)
if getNthPolygonalNumber(guess, 5) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' pentagonal number')
# hexagonal
guess = findPolygonalNumber(n, 6)
if getNthPolygonalNumber(guess, 6) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' hexagonal number')
# heptagonal
guess = findPolygonalNumber(n, 7)
if getNthPolygonalNumber(guess, 7) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' heptagonal number')
# octagonal
guess = findPolygonalNumber(n, 8)
if getNthPolygonalNumber(guess, 8) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' octagonal number')
# nonagonal
guess = findPolygonalNumber(n, 9)
if getNthPolygonalNumber(guess, 9) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' nonagonal number')
# decagonal
guess = findPolygonalNumber(n, 10)
if getNthPolygonalNumber(guess, 10) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' decagonal number')
# if n > 1:
# for i in range( 11, 101 ):
# if getNthPolygonalNumber( findPolygonalNumber( n, i ), i ) == n:
# print( indent + str( i ) + '-gonal' )
# centered triangular
guess = findCenteredPolygonalNumber(n, 3)
if getNthCenteredPolygonalNumber(guess, 3) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered triangular')
# centered square
guess = findCenteredPolygonalNumber(n, 4)
if getNthCenteredPolygonalNumber(guess, 4) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered square number')
# centered pentagonal
guess = findCenteredPolygonalNumber(n, 5)
if getNthCenteredPolygonalNumber(guess, 5) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered pentagonal number')
# centered hexagonal
guess = findCenteredPolygonalNumber(n, 6)
if getNthCenteredPolygonalNumber(guess, 6) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered hexagonal number')
# centered heptagonal
guess = findCenteredPolygonalNumber(n, 7)
if getNthCenteredPolygonalNumber(guess, 7) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered heptagonal number')
# centered octagonal
guess = findCenteredPolygonalNumber(n, 8)
if getNthCenteredPolygonalNumber(guess, 8) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered octagonal number')
# centered nonagonal
guess = findCenteredPolygonalNumber(n, 9)
if getNthCenteredPolygonalNumber(guess, 9) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered nonagonal number')
# centered decagonal
guess = findCenteredPolygonalNumber(n, 10)
if getNthCenteredPolygonalNumber(guess, 10) == n:
print(indent + 'the ' + getShortOrdinalName(guess) +
' centered decagonal number')
# pandigital
if isPandigital(n):
print(indent + 'pandigital')
# for i in range( 4, 21 ):
# if isBaseKPandigital( n, i ):
# print( indent + 'base ' + str( i ) + ' pandigital' )
# Fibonacci
result = findInput(n, fib, lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Fibonacci number')
# Tribonacci
result = findInput(n, lambda n: getNthKFibonacciNumber(n, 3),
lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Tribonacci number')
# Tetranacci
result = findInput(n, lambda n: getNthKFibonacciNumber(n, 4),
lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Tetranacci number')
# Pentanacci
result = findInput(n, lambda n: getNthKFibonacciNumber(n, 5),
lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Pentanacci number')
# Hexanacci
result = findInput(n, lambda n: getNthKFibonacciNumber(n, 6),
lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Hexanacci number')
# Heptanacci
result = findInput(n, lambda n: getNthKFibonacciNumber(n, 7),
lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Heptanacci number')
# Octanacci
result = findInput(n, lambda n: getNthKFibonacciNumber(n, 8),
lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Octanacci number')
# Lucas numbers
result = findInput(n, getNthLucasNumber, lambda n: fmul(log10(n), 5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Lucas number')
# base-k repunits
if n > 1:
for i in range(2, 21):
result = findInput(n,
lambda x: getNthBaseKRepunit(x, i),
lambda n: log(n, i),
minimum=1)
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' base-' + str(i) + ' repunit')
# Jacobsthal numbers
result = findInput(n, getNthJacobsthalNumber, lambda n: fmul(log(n), 1.6))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Jacobsthal number')
# Padovan numbers
result = findInput(n, getNthPadovanNumber, lambda n: fmul(log10(n), 9))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Padovan number')
# Fibonorial numbers
result = findInput(n, getNthFibonorial, lambda n: sqrt(fmul(log(n), 10)))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Fibonorial number')
# Mersenne primes
result = findInput(n, getNthMersennePrime,
lambda n: fadd(fmul(log(log(sqrt(n))), 2.7), 3))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Mersenne prime')
# perfect number
result = findInput(n, getNthPerfectNumber,
lambda n: fmul(log(log(n)), 2.6))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' perfect number')
# Mersenne exponent
result = findInput(n,
getNthMersenneExponent,
lambda n: fmul(log(n), 2.7),
maximum=50)
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Mersenne exponent')
if not isPrimeFlag and n != 1 and n <= LARGEST_NUMBER_TO_FACTOR:
# deficient
if isDeficient(n):
print(indent + 'deficient')
# abundant
if isAbundant(n):
print(indent + 'abundant')
# k_hyperperfect
for i in sorted(list(set(sorted(
downloadOEISSequence(34898)[:500]))))[1:]:
if i > n:
break
if isKHyperperfect(n, i):
print(indent + str(i) + '-hyperperfect')
break
# smooth
for i in getPrimes(2, 50):
if isSmooth(n, i):
print(indent + str(i) + '-smooth')
break
# rough
previousPrime = 2
for i in getPrimes(2, 50):
if not isRough(n, i):
print(indent + str(previousPrime) + '-rough')
break
previousPrime = i
# is_semiprime
if isSemiprime(n):
print(indent + 'semiprime')
# is_sphenic
elif isSphenic(n):
print(indent + 'sphenic')
elif isSquareFree(n):
print(indent + 'square-free')
# powerful
if isPowerful(n):
print(indent + 'powerful')
# factorial
result = findInput(n, getNthFactorial, lambda n: power(log10(n), 0.92))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' factorial number')
# alternating factorial
result = findInput(n, getNthAlternatingFactorial,
lambda n: fmul(sqrt(log(n)), 0.72))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' alternating factorial number')
# double factorial
if n == 1:
result = (True, 1)
else:
result = findInput(n, getNthDoubleFactorial,
lambda n: fdiv(power(log(log(n)), 4), 7))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' double factorial number')
# hyperfactorial
result = findInput(n, getNthHyperfactorial,
lambda n: fmul(sqrt(log(n)), 0.8))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' hyperfactorial number')
# subfactorial
result = findInput(n, getNthSubfactorial, lambda n: fmul(log10(n), 1.1))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' subfactorial number')
# superfactorial
if n == 1:
result = (True, 1)
else:
result = findInput(n, getNthSuperfactorial,
lambda n: fadd(sqrt(log(n)), 1))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' superfactorial number')
# pernicious
if isPernicious(n):
print(indent + 'pernicious')
# pronic
if isPronic(n):
print(indent + 'pronic')
# Achilles
if isAchillesNumber(n):
print(indent + 'an Achilles number')
# antiharmonic
if isAntiharmonic(n):
print(indent + 'antiharmonic')
# unusual
if isUnusual(n):
print(indent + 'unusual')
# hyperperfect
for i in range(2, 21):
if isKHyperperfect(n, i):
print(indent + str(i) + '-hyperperfect')
# Ruth-Aaron
if isRuthAaronNumber(n):
print(indent + 'a Ruth-Aaron number')
# Smith numbers
if isSmithNumber(n):
print(indent + 'a Smith number')
# base-k Smith numbers
for i in range(2, 10):
if isBaseKSmithNumber(n, i):
print(indent + 'a base-' + str(i) + ' Smith number')
# order-k Smith numbers
for i in range(2, 11):
if isOrderKSmithNumber(n, i):
print(indent + 'an order-' + str(i) + ' Smith number')
# polydivisible
if isPolydivisible(n):
print(indent + 'polydivisible')
# Carol numbers
result = findInput(n, getNthCarolNumber,
lambda n: fmul(log10(n), fdiv(5, 3)))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Carol number')
# Kynea numbers
result = findInput(n, getNthKyneaNumber,
lambda n: fmul(log10(n), fdiv(5, 3)))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Kynea number')
# Leonardo numbers
result = findInput(n, getNthLeonardoNumber,
lambda n: fsub(log(n, phi), fdiv(1, phi)))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Leonardo number')
# Thabit numbers
result = findInput(n, getNthThabitNumber, lambda n: fmul(log10(n), 3.25))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Thabit number')
# Carmmichael
if isCarmichaelNumber(n):
print(indent + 'a Carmichael number')
# narcissistic
if isNarcissistic(n):
print(indent + 'narcissistic')
# PDI
if isPerfectDigitalInvariant(n):
print(indent + 'a perfect digital invariant')
# PDDI
if isPerfectDigitToDigitInvariant(n, 10):
print(indent + 'a perfect digit-to-digit invariant in base 10')
# Kaprekar
if isKaprekarNumber(n):
print(indent + 'a Kaprekar number')
# automorphic
if isAutomorphic(n):
print(indent + 'automorphic')
# trimorphic
if isTrimorphic(n):
print(indent + 'trimorphic')
# k-morphic
for i in range(4, 11):
if isKMorphic(n, i):
print(indent + str(i) + '-morphic')
# bouncy
if isBouncy(n):
print(indent + 'bouncy')
# step number
if isStepNumber(n):
print(indent + 'a step number')
# Apery numbers
result = findInput(n, getNthAperyNumber,
lambda n: fadd(fdiv(log(n), 1.5), 1))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Apery number')
# Delannoy numbers
result = findInput(n, getNthDelannoyNumber, lambda n: fmul(log10(n), 1.35))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Delannoy number')
# Schroeder numbers
result = findInput(n, getNthSchroederNumber, lambda n: fmul(log10(n), 1.6))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Schroeder number')
# Schroeder-Hipparchus numbers
result = findInput(n, getNthSchroederHipparchusNumber,
lambda n: fdiv(log10(n), 1.5))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Schroeder-Hipparchus number')
# Motzkin numbers
result = findInput(n, getNthMotzkinNumber, log)
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Motzkin number')
# Pell numbers
result = findInput(n, getNthPellNumber, lambda n: fmul(log(n), 1.2))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Pell number')
# Sylvester numbers
if n > 1:
result = findInput(n, getNthSylvesterNumber,
lambda n: sqrt(sqrt(log(n))))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Sylvester number')
# partition numbers
result = findInput(n, getPartitionNumber, lambda n: power(log(n), 1.56))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' partition number')
# menage numbers
result = findInput(n, getNthMenageNumber, lambda n: fdiv(log10(n), 1.2))
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' menage number')
print()
print(int(n), 'has:')
# number of digits
digits = log10(n)
if isInteger(digits):
digits += 1
else:
digits = ceil(digits)
print(indent + str(int(digits)) + ' digit' + ('s' if digits > 1 else ''))
# digit sum
print(indent + 'a digit sum of ' + str(int(sumDigits(n))))
# digit product
digitProduct = multiplyDigits(n)
print(indent + 'a digit product of ' + str(int(digitProduct)))
# non-zero digit product
if digitProduct == 0:
print(indent + 'a non-zero digit product of ' +
str(int(multiplyNonzeroDigits(n))))
if not isPrimeFlag and n != 1 and n <= LARGEST_NUMBER_TO_FACTOR:
# factors
factors = getFactors(n)
factorCount = len(factors)
print(indent + str(factorCount) + ' prime factor' +
('s' if factorCount > 1 else '') + ': ' +
', '.join([str(int(i)) for i in factors]))
# number of divisors
divisorCount = int(getDivisorCount(n))
print(indent + str(divisorCount) + ' divisor' +
('s' if divisorCount > 1 else ''))
if n <= LARGEST_NUMBER_TO_FACTOR:
print(indent + 'a divisor sum of ' + str(int(getSigma(n))))
print(indent + 'a Stern value of ' + str(int(getNthSternNumber(n))))
calkinWilf = getNthCalkinWilf(n)
print(indent + 'a Calkin-Wilf value of ' + str(int(calkinWilf[0])) +
'/' + str(int(calkinWilf[1])))
print(indent + 'a Mobius value of ' + str(int(getNthMobiusNumber(n))))
print(indent + 'a radical of ' + str(int(getRadical(n))))
print(indent + 'a Euler phi value of ' + str(int(getEulerPhi(n))))
print(indent + 'a digital root of ' + str(int(getDigitalRoot(n))))
if hasUniqueDigits(n):
print(indent + 'unique digits')
print(indent + 'a multiplicative persistence of ' +
str(int(getPersistence(n))))
print(indent + 'an Erdos persistence of ' +
str(int(getErdosPersistence(n))))
if n > 9 and isIncreasing(n):
if not isDecreasing(n):
print(indent + 'increasing digits')
elif n > 9 and isDecreasing(n):
print(indent + 'decreasing digits')
print()
return n
def getSuperRoot( n, k ):
k = fsub( real_int( k ), 1 )
value = fmul( k, log( n ) )
return root( fdiv( value, lambertw( value ) ), k )
trigUnit = TrigUnit.Radians
Backend.engineFunction("=", lambda op: mpmath.mpmathify(op), operands=1)
Backend.engineFunction("+", lambda op: op[0] + op[1])
Backend.engineFunction("-", lambda op: op[0] - op[1])
Backend.engineFunction("mul", lambda op: op[0] * op[1])
Backend.engineFunction("div", lambda op: op[0] / op[1])
Backend.engineFunction("^", lambda op: op[0] ** op[1])
Backend.engineFunction("10^x", lambda op: 10 ** op, operands=1)
Backend.engineFunction("x^2", lambda op: op ** 2, operands=1)
Backend.engineFunction("neg", lambda op: op * -1, operands=1)
Backend.engineFunction("%", lambda op: op[0] * op[1] / 100)
Backend.engineFunction("inv", lambda op: 1 / op, operands=1)
Backend.engineFunction("sqrt", lambda op: mpmath.sqrt(op), operands=1)
Backend.engineFunction("nthroot", lambda op: mpmath.root(op[0], op[1]))
Backend.engineFunction("log", lambda op: mpmath.log(op, b=10), operands=1)
Backend.engineFunction("ln", lambda op: mpmath.log(op), operands=1)
Backend.engineFunction("e^x", lambda op: mpmath.exp(op), operands=1)
Backend.engineFunction("factorial", lambda op: mpmath.factorial(op), operands=1)
@Backend.engineFunction(operands=-1)
def addall(op):
expr = op[0]
for i in range(1, len(op)):
expr = expr + op[i]
return expr
@Backend.engineFunction(operands=-1)
def suball(op):
expr = op[0]
def describeInteger( n ):
if n < 1:
raise ValueError( "'describe' requires a positive integer argument" )
indent = ' ' * 4
print( )
print( real_int( n ), 'is:' )
if isOdd( n ):
print( indent + 'odd' )
elif isEven( n ):
print( indent + 'even' )
if isPrimeNumber( n ):
isPrime = True
print( indent + 'prime' )
elif n > 3:
isPrime = False
print( indent + 'composite' )
else:
isPrime = False
if isKthPower( n, 2 ):
print( indent + 'the ' + getShortOrdinalName( sqrt( n ) ) + ' square number' )
if isKthPower( n, 3 ):
print( indent + 'the ' + getShortOrdinalName( cbrt( n ) ) + ' cube number' )
for i in arange( 4, fadd( ceil( log( fabs( n ), 2 ) ), 1 ) ):
if isKthPower( n, i ):
print( indent + 'the ' + getShortOrdinalName( root( n, i ) ) + ' ' + \
getNumberName( i, True ) + ' power' )
# triangular
guess = findPolygonalNumber( n, 3 )
if getNthPolygonalNumber( guess, 3 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' triangular number' )
# pentagonal
guess = findPolygonalNumber( n, 5 )
if getNthPolygonalNumber( guess, 5 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' pentagonal number' )
# hexagonal
guess = findPolygonalNumber( n, 6 )
if getNthPolygonalNumber( guess, 6 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' hexagonal number' )
# heptagonal
guess = findPolygonalNumber( n, 7 )
if getNthPolygonalNumber( guess, 7 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' heptagonal number' )
# octagonal
guess = findPolygonalNumber( n, 8 )
if getNthPolygonalNumber( guess, 8 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' octagonal number' )
# nonagonal
guess = findPolygonalNumber( n, 9 )
if getNthPolygonalNumber( guess, 9 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' nonagonal number' )
# decagonal
guess = findPolygonalNumber( n, 10 )
if getNthPolygonalNumber( guess, 10 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' decagonal number' )
#if n > 1:
# for i in range( 11, 101 ):
# if getNthPolygonalNumber( findPolygonalNumber( n, i ), i ) == n:
# print( indent + str( i ) + '-gonal' )
# centered triangular
guess = findCenteredPolygonalNumber( n, 3 )
if getNthCenteredPolygonalNumber( guess, 3 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered triangular' )
# centered square
guess = findCenteredPolygonalNumber( n, 4 )
if getNthCenteredPolygonalNumber( guess, 4 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered square number' )
# centered pentagonal
guess = findCenteredPolygonalNumber( n, 5 )
if getNthCenteredPolygonalNumber( guess, 5 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered pentagonal number' )
# centered hexagonal
guess = findCenteredPolygonalNumber( n, 6 )
if getNthCenteredPolygonalNumber( guess, 6 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered hexagonal number' )
# centered heptagonal
guess = findCenteredPolygonalNumber( n, 7 )
if getNthCenteredPolygonalNumber( guess, 7 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered heptagonal number' )
# centered octagonal
guess = findCenteredPolygonalNumber( n, 8 )
if getNthCenteredPolygonalNumber( guess, 8 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered octagonal number' )
# centered nonagonal
guess = findCenteredPolygonalNumber( n, 9 )
if getNthCenteredPolygonalNumber( guess, 9 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered nonagonal number' )
# centered decagonal
guess - findCenteredPolygonalNumber( n, 10 )
if getNthCenteredPolygonalNumber( guess, 10 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered decagonal number' )
# pandigital
if isPandigital( n ):
print( indent + 'pandigital' )
#for i in range( 4, 21 ):
# if isBaseKPandigital( n, i ):
# print( indent + 'base ' + str( i ) + ' pandigital' )
# Fibonacci
result = findInput( n, fib, lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Fibonacci number' )
# Tribonacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 3 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Tribonacci number' )
# Tetranacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 4 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Tetranacci number' )
# Pentanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 5 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Pentanacci number' )
# Hexanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 6 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Hexanacci number' )
# Heptanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 7 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Heptanacci number' )
# Octanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 8 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Octanacci number' )
# Lucas numbers
result = findInput( n, getNthLucasNumber, lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Lucas number' )
# base-k repunits
if n > 1:
for i in range( 2, 21 ):
result = findInput( n, lambda x: getNthBaseKRepunit( x, i ), lambda n: log( n, i ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' base-' + str( i ) + ' repunit' )
# Jacobsthal numbers
result = findInput( n, getNthJacobsthalNumber, lambda n: fmul( log( n ), 1.6 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Jacobsthal number' )
# Padovan numbers
result = findInput( n, getNthPadovanNumber, lambda n: fmul( log10( n ), 9 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Padovan number' )
# Fibonorial numbers
result = findInput( n, getNthFibonorial, lambda n: sqrt( fmul( log( n ), 10 ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Fibonorial number' )
# Mersenne primes
result = findInput( n, getNthMersennePrime, lambda n: fadd( fmul( log( log( sqrt( n ) ) ), 2.7 ), 3 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Mersenne prime' )
## perfect number
result = findInput( n, getNthPerfectNumber, lambda n: fmul( log( log( n ) ), 2.6 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' perfect number' )
# Mersenne exponent
result = findInput( n, getNthMersenneExponent, lambda n: fmul( log( n ), 2.7 ), max=50 )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Mersenne exponent' )
if not isPrime and n != 1 and n <= largestNumberToFactor:
# deficient
if isDeficient( n ):
print( indent + 'deficient' )
# abundant
if isAbundant( n ):
print( indent + 'abundant' )
# k_hyperperfect
for i in sorted( list( set( sorted( downloadOEISSequence( 34898 )[ : 500 ] ) ) ) )[ 1 : ]:
if i > n:
break
if isKHyperperfect( n, i ):
print( indent + str( i ) + '-hyperperfect' )
break
# smooth
for i in getPrimes( 2, 50 ):
if isSmooth( n, i ):
print( indent + str( i ) + '-smooth' )
break
# rough
previousPrime = 2
for i in getPrimes( 2, 50 ):
if not isRough( n, i ):
print( indent + str( previousPrime ) + '-rough' )
break
previousPrime = i
# is_semiprime
if isSemiprime( n ):
print( indent + 'semiprime' )
# is_sphenic
elif isSphenic( n ):
print( indent + 'sphenic' )
elif isSquareFree( n ):
print( indent + 'square-free' )
# powerful
if isPowerful( n ):
print( indent + 'powerful' )
# factorial
result = findInput( n, getNthFactorial, lambda n: power( log10( n ), 0.92 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' factorial number' )
# alternating factorial
result = findInput( n, getNthAlternatingFactorial, lambda n: fmul( sqrt( log( n ) ), 0.72 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' alternating factorial number' )
# double factorial
if n == 1:
result = ( True, 1 )
else:
result = findInput( n, getNthDoubleFactorial, lambda n: fdiv( power( log( log( n ) ), 4 ), 7 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' double factorial number' )
# hyperfactorial
result = findInput( n, getNthHyperfactorial, lambda n: fmul( sqrt( log( n ) ), 0.8 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' hyperfactorial number' )
# subfactorial
result = findInput( n, getNthSubfactorial, lambda n: fmul( log10( n ), 1.1 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' subfactorial number' )
# superfactorial
if n == 1:
result = ( True, 1 )
else:
result = findInput( n, getNthSuperfactorial, lambda n: fadd( sqrt( log( n ) ), 1 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' superfactorial number' )
# pernicious
if isPernicious( n ):
print( indent + 'pernicious' )
# pronic
if isPronic( n ):
print( indent + 'pronic' )
# Achilles
if isAchillesNumber( n ):
print( indent + 'an Achilles number' )
# antiharmonic
if isAntiharmonic( n ):
print( indent + 'antiharmonic' )
# unusual
if isUnusual( n ):
print( indent + 'unusual' )
# hyperperfect
for i in range( 2, 21 ):
if isKHyperperfect( n, i ):
print( indent + str( i ) + '-hyperperfect' )
# Ruth-Aaron
if isRuthAaronNumber( n ):
print( indent + 'a Ruth-Aaron number' )
# Smith numbers
if isSmithNumber( n ):
print( indent + 'a Smith number' )
# base-k Smith numbers
for i in range( 2, 10 ):
if isBaseKSmithNumber( n, i ):
print( indent + 'a base-' + str( i ) + ' Smith number' )
# order-k Smith numbers
for i in range( 2, 11 ):
if isOrderKSmithNumber( n, i ):
print( indent + 'an order-' + str( i ) + ' Smith number' )
# polydivisible
if isPolydivisible( n ):
print( indent + 'polydivisible' )
# Carol numbers
result = findInput( n, getNthCarolNumber, lambda n: fmul( log10( n ), fdiv( 5, 3 ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Carol number' )
# Kynea numbers
result = findInput( n, getNthKyneaNumber, lambda n: fmul( log10( n ), fdiv( 5, 3 ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Kynea number' )
# Leonardo numbers
result = findInput( n, getNthLeonardoNumber, lambda n: fsub( log( n, phi ), fdiv( 1, phi ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Leonardo number' )
# Riesel numbers
result = findInput( n, getNthRieselNumber, lambda n: fmul( log( n ), 1.25 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Riesel number' )
# Thabit numbers
result = findInput( n, getNthThabitNumber, lambda n: fmul( log10( n ), 3.25 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Thabit number' )
# Carmmichael
if isCarmichaelNumber( n ):
print( indent + 'a Carmichael number' )
# narcissistic
if isNarcissistic( n ):
print( indent + 'narcissistic' )
# PDI
if isPerfectDigitalInvariant( n ):
print( indent + 'a perfect digital invariant' )
# PDDI
if isPerfectDigitToDigitInvariant( n, 10 ):
print( indent + 'a perfect digit-to-digit invariant in base 10' )
# Kaprekar
if isKaprekar( n ):
print( indent + 'a Kaprekar number' )
# automorphic
if isAutomorphic( n ):
print( indent + 'automorphic' )
# trimorphic
if isTrimorphic( n ):
print( indent + 'trimorphic' )
# k-morphic
for i in range( 4, 11 ):
if isKMorphic( n, i ):
print( indent + str( i ) + '-morphic' )
# bouncy
if isBouncy( n ):
print( indent + 'bouncy' )
# step number
if isStepNumber( n ):
print( indent + 'a step number' )
# Apery numbers
result = findInput( n, getNthAperyNumber, lambda n: fadd( fdiv( log( n ), 1.5 ), 1 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Apery number' )
# Delannoy numbers
result = findInput( n, getNthDelannoyNumber, lambda n: fmul( log10( n ), 1.35 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Delannoy number' )
# Schroeder numbers
result = findInput( n, getNthSchroederNumber, lambda n: fmul( log10( n ), 1.6 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Schroeder number' )
# Schroeder-Hipparchus numbers
result = findInput( n, getNthSchroederHipparchusNumber, lambda n: fdiv( log10( n ), 1.5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Schroeder-Hipparchus number' )
# Motzkin numbers
result = findInput( n, getNthMotzkinNumber, lambda n: log( n ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Motzkin number' )
# Pell numbers
result = findInput( n, getNthPellNumber, lambda n: fmul( log( n ), 1.2 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Pell number' )
# Sylvester numbers
if n > 1:
result = findInput( n, getNthSylvesterNumber, lambda n: sqrt( sqrt( log( n ) ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Sylvester number' )
# partition numbers
result = findInput( n, getPartitionNumber, lambda n: power( log( n ), 1.56 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' partition number' )
# menage numbers
result = findInput( n, getNthMenageNumber, lambda n: fdiv( log10( n ), 1.2 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' menage number' )
print( )
print( int( n ), 'has:' )
# number of digits
digits = log10( n )
if isInteger( digits ):
digits += 1
else:
digits = ceil( digits )
print( indent + str( int( digits ) ) + ' digit' + ( 's' if digits > 1 else '' ) )
# digit sum
print( indent + 'a digit sum of ' + str( int( sumDigits( n ) ) ) )
# digit product
digitProduct = multiplyDigits( n )
print( indent + 'a digit product of ' + str( int( digitProduct ) ) )
# non-zero digit product
if digitProduct == 0:
print( indent + 'a non-zero digit product of ' + str( int( multiplyNonzeroDigits( n ) ) ) )
if not isPrime and n != 1 and n <= largestNumberToFactor:
# factors
factors = getFactors( n )
factorCount = len( factors )
print( indent + str( factorCount ) + ' prime factor' + ( 's' if factorCount > 1 else '' ) + \
': ' + ', '.join( [ str( int( i ) ) for i in factors ] ) )
# number of divisors
divisorCount = int( getDivisorCount( n ) )
print( indent + str( divisorCount ) + ' divisor' + ( 's' if divisorCount > 1 else '' ) )
if n <= largestNumberToFactor:
print( indent + 'a sum of divisors of ' + str( int( getSigma( n ) ) ) )
print( indent + 'a Stern value of ' + str( int( getNthStern( n ) ) ) )
calkin_wilf = getNthCalkinWilf( n )
print( indent + 'a Calkin-Wilf value of ' + str( int( calkin_wilf[ 0 ] ) ) + '/' + str( int( calkin_wilf[ 1 ] ) ) )
print( indent + 'a Mobius value of ' + str( int( getMobius( n ) ) ) )
print( indent + 'a radical of ' + str( int( getRadical( n ) ) ) )
print( indent + 'a Euler phi value of ' + str( int( getEulerPhi( n ) ) ) )
print( indent + 'a digital root of ' + str( int( getDigitalRoot( n ) ) ) )
if hasUniqueDigits( n ):
print( indent + 'unique digits' )
print( indent + 'a multiplicative persistence of ' + str( int( getPersistence( n ) ) ) )
print( indent + 'an Erdos persistence of ' + str( int( getErdosPersistence( n ) ) ) )
if n > 9 and isIncreasing( n ):
if not isDecreasing( n ):
print( indent + 'increasing digits' )
elif n > 9 and isDecreasing( n ):
print( indent + 'decreasing digits' )
print( )
return n
def getSuperRootOperator(n, k):
'''Returns the positive, real kth super root of n.'''
k = fsub(k, 1)
value = fmul(k, log(n))
return root(fdiv(value, lambertw(value)), k)
def getSuperRoots( n, k ):
k = fsub( real_int( k ), 1 )
factors = [ fmul( i, root( k, k ) ) for i in unitroots( int( k ) ) ]
base = root( fdiv( log( n ), lambertw( fmul( k, log( n ) ) ) ), k )
return [ fmul( i, base ) for i in factors ]
def solveCubicPolynomial( a, b, c, d ):
if mp.dps < 50:
mp.dps = 50
if a == 0:
return solveQuadraticPolynomial( b, c, d )
f = fdiv( fsub( fdiv( fmul( 3, c ), a ), fdiv( power( b, 2 ), power( a, 2 ) ) ), 3 )
g = fdiv( fadd( fsub( fdiv( fmul( 2, power( b, 3 ) ), power( a, 3 ) ),
fdiv( fprod( [ 9, b, c ] ), power( a, 2 ) ) ),
fdiv( fmul( 27, d ), a ) ), 27 )
h = fadd( fdiv( power( g, 2 ), 4 ), fdiv( power( f, 3 ), 27 ) )
# all three roots are the same
if h == 0:
x1 = fneg( root( fdiv( d, a ), 3 ) )
x2 = x1
x3 = x2
# two imaginary and one real root
elif h > 0:
r = fadd( fneg( fdiv( g, 2 ) ), sqrt( h ) )
if r < 0:
s = fneg( root( fneg( r ), 3 ) )
else:
s = root( r, 3 )
t = fsub( fneg( fdiv( g, 2 ) ), sqrt( h ) )
if t < 0:
u = fneg( root( fneg( t ), 3 ) )
else:
u = root( t, 3 )
x1 = fsub( fadd( s, u ), fdiv( b, fmul( 3, a ) ) )
real = fsub( fdiv( fneg( fadd( s, u ) ), 2 ), fdiv( b, fmul( 3, a ) ) )
imaginary = fdiv( fmul( fsub( s, u ), sqrt( 3 ) ), 2 )
x2 = mpc( real, imaginary )
x3 = mpc( real, fneg( imaginary ) )
# all real roots
else:
j = sqrt( fsub( fdiv( power( g, 2 ), 4 ), h ) )
k = acos( fneg( fdiv( g, fmul( 2, j ) ) ) )
if j < 0:
l = fneg( root( fneg( j ), 3 ) )
else:
l = root( j, 3 )
m = cos( fdiv( k, 3 ) )
n = fmul( sqrt( 3 ), sin( fdiv( k, 3 ) ) )
p = fneg( fdiv( b, fmul( 3, a ) ) )
x1 = fsub( fmul( fmul( 2, l ), cos( fdiv( k, 3 ) ) ), fdiv( b, fmul( 3, a ) ) )
x2 = fadd( fmul( fneg( l ), fadd( m, n ) ), p )
x3 = fadd( fmul( fneg( l ), fsub( m, n ) ), p )
return [ chop( x1 ), chop( x2 ), chop( x3 ) ]
'pi':
mp.pi,
'degree':
mp.degree, #pi / 180
'e':
mp.e,
'phi':
mp.phi, #golden ratio = 1.61803...
'euler':
mp.euler, #euler's constance = 0.577216...
'mpf': ['primitive', [lambda x, y: mp.mpf(str(x)), None]],
'mpc': ['primitive', [lambda x, y: mp.mpc(x, y[0]), None]],
#
'sqrt': ['primitive', [lambda x, y: mp.sqrt(x), None]],
'cbrt': ['primitive', [lambda x, y: mp.cbrt(x), None]],
'root': ['primitive', [lambda x, y: mp.root(x, y[0]), None]], # y's root
'unitroots': ['primitive', [lambda x, y: Vector(mp.unitroots(x)),
None]], #
'hypot': ['primitive', [lambda x, y: mp.hypot(x, y[0]),
None]], # sqrt(x**2+y**2)
#
'sin': ['primitive', [lambda x, y: mp.sin(x), None]],
'cos': ['primitive', [lambda x, y: mp.cos(x), None]],
'tan': ['primitive', [lambda x, y: mp.tan(x), None]],
'sinpi': ['primitive', [lambda x, y: mp.sinpi(x), None]], #sin(x * pi)
'cospi': ['primitive', [lambda x, y: mp.cospi(x), None]],
'sec': ['primitive', [lambda x, y: mp.sec(x), None]],
'csc': ['primitive', [lambda x, y: mp.csc(x), None]],
'cot': ['primitive', [lambda x, y: mp.cot(x), None]],
'asin': ['primitive', [lambda x, y: mp.asin(x), None]],
'acos': ['primitive', [lambda x, y: mp.acos(x), None]],
def solveQuarticPolynomial( _a, _b, _c, _d, _e ):
if mp.dps < 50:
mp.dps = 50
# maybe it's really an order-3 polynomial
if _a == 0:
return solveCubicPolynomial( _b, _c, _d, _e )
# degenerate case, just return the two real and two imaginary 4th roots of the
# constant term divided by the 4th root of a
elif _b == 0 and _c == 0 and _d == 0:
e = fdiv( _e, _a )
f = root( _a, 4 )
x1 = fdiv( root( fneg( e ), 4 ), f )
x2 = fdiv( fneg( root( fneg( e ), 4 ) ), f )
x3 = fdiv( mpc( 0, root( fneg( e ), 4 ) ), f )
x4 = fdiv( mpc( 0, fneg( root( fneg( e ), 4 ) ) ), f )
return [ x1, x2, x3, x4 ]
# otherwise we have a regular quartic to solve
b = fdiv( _b, _a )
c = fdiv( _c, _a )
d = fdiv( _d, _a )
e = fdiv( _e, _a )
# we turn the equation into a cubic that we can solve
f = fsub( c, fdiv( fmul( 3, power( b, 2 ) ), 8 ) )
g = fsum( [ d, fdiv( power( b, 3 ), 8 ), fneg( fdiv( fmul( b, c ), 2 ) ) ] )
h = fsum( [ e, fneg( fdiv( fmul( 3, power( b, 4 ) ), 256 ) ),
fmul( power( b, 2 ), fdiv( c, 16 ) ), fneg( fdiv( fmul( b, d ), 4 ) ) ] )
y1, y2, y3 = solveCubicPolynomial( 1, fdiv( f, 2 ), fdiv( fsub( power( f, 2 ), fmul( 4, h ) ), 16 ),
fneg( fdiv( power( g, 2 ), 64 ) ) )
# pick two non-zero roots, if there are two imaginary roots, use them
if y1 == 0:
root1 = y2
root2 = y3
elif y2 == 0:
root1 = y1
root2 = y3
elif y3 == 0:
root1 = y1
root2 = y2
elif im( y1 ) != 0:
root1 = y1
if im( y2 ) != 0:
root2 = y2
else:
root2 = y3
else:
root1 = y2
root2 = y3
# more variables...
p = sqrt( root1 )
q = sqrt( root2 )
r = fdiv( fneg( g ), fprod( [ 8, p, q ] ) )
s = fneg( fdiv( b, 4 ) )
# put together the 4 roots
x1 = fsum( [ p, q, r, s ] )
x2 = fsum( [ p, fneg( q ), fneg( r ), s ] )
x3 = fsum( [ fneg( p ), q, fneg( r ), s ] )
x4 = fsum( [ fneg( p ), fneg( q ), r, s ] )
return [ chop( x1 ), chop( x2 ), chop( x3 ), chop( x4 ) ]
