def isAchillesNumber( n ):
factors = getECMFactors( n ) if g.ecm else getFactors( n )
if min( [ i[ 1 ] for i in factors ] ) < 2:
return 0
return 1 if getGCD( [ i[ 1 ] for i in factors ] ) == 1 else 0
def getSigma( target ):
'''
Returns the sum of the divisors of n, including 1 and n.
http://math.stackexchange.com/questions/22721/is-there-a-formula-to-calculate-the-sum-of-all-proper-divisors-of-a-number
'''
n = floor( target )
if real( n ) == 0:
return 0
elif n == 1:
return 1
factors = getECMFactors( n ) if g.ecm else getFactors( n )
result = 1
for factor in factors:
numerator = fsub( power( factor[ 0 ], fadd( factor[ 1 ], 1 ) ), 1 )
denominator = fsub( factor[ 0 ], 1 )
#debugPrint( 'sigma', numerator, denominator )
result = fmul( result, fdiv( numerator, denominator ) )
if result != floor( result ):
raise ValueError( 'insufficient precision for \'sigma\', increase precision (-p))' )
return result
def isUnusual( n ):
if real_int( n ) < 2:
return 0
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return 1 if max( [ i[ 0 ] for i in factors ] ) > sqrt( n ) else 0
def isSquareFree( n ):
if real_int( n ) == 0:
return 0
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return 1 if max( [ i[ 1 ] for i in factors ] ) == 1 else 0
def isSphenic( n ):
factors = getECMFactors( n ) if g.ecm else getFactors( n )
if len( factors ) != 3:
return 0
return 1 if max( [ i[ 1 ] for i in factors ] ) == 1 else 0
def isRough( n, k ):
if real( n ) < real( k ):
return 0
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return 1 if min( [ i[ 0 ] for i in factors ] ) >= k else 0
def isSmooth( n, k ):
if real( n ) < real( k ):
return 0
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return 1 if max( [ i[ 0 ] for i in factors ] ) <= k else 0
def getDivisors( n ):
if n == 0:
return [ 0 ]
elif n == 1:
return [ 1 ]
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return sorted( createDivisorList( [ ], factors ) )
def getMobius( n ):
if real( n ) == 1:
return 1
factors = getECMFactors( n ) if g.ecm else getFactors( n )
for i in factors:
if i[ 1 ] > 1:
return 0
if len( factors ) % 2:
return -1
else:
return 1
def getSigmaN( n, k ):
'''
Returns the sum of the divisors of n, including 1 and n, to the k power.
https://oeis.org/A001157
'''
if real( n ) == 0:
return 0
elif n == 1:
return 1
factors = getECMFactors( n ) if g.ecm else getFactors( n )
result = 1
for factor in factors:
numerator = fsub( power( factor[ 0 ], fmul( fadd( factor[ 1 ], 1 ), k ) ), 1 )
denominator = fsub( power( factor[ 0 ], k ), 1 )
result = fmul( result, fdiv( numerator, denominator ) )
if result != floor( result ):
raise ValueError( 'insufficient precision for \'sigma_n\', increase precision (-p))' )
return result
def getEulerPhi( n ):
if real( n ) < 2:
return n
if g.ecm:
return reduce( fmul, ( fmul( fsub( i[ 0 ], 1 ), power( i[ 0 ], fsub( i[ 1 ], 1 ) ) ) for i in getECMFactors( n ) ) )
else:
return reduce( fmul, ( fmul( fsub( i[ 0 ], 1 ), power( i[ 0 ], fsub( i[ 1 ], 1 ) ) ) for i in getFactors( n ) ) )
def getDivisorCount( n ):
if n == 1:
return 1
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return fprod( [ i[ 1 ] + 1 for i in factors ] )
def isPowerful( n ):
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return 1 if min( [ i[ 1 ] for i in factors ] ) >= 2 else 0
def isKSemiPrime( n, k ):
factors = getECMFactors( n ) if g.ecm else getFactors( n )
return 1 if sum( [ i[ 1 ] for i in factors ] ) == k else 0