예제 #1
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def impfact(n, primelist = None, primefactorization = None):
	"""returns all of the factors needed to test whether a
number is primitive abundant"""
	if primefactorization == None:
		primefactorization = primes2.pf(n, primelist)
	r = []
	for x in primefactorization:
		r.append(n/x[0])
	return r
예제 #2
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def tau(n, primelist = None, primefactorization = None):
	'''the number-theoretic method for computing the number of factors of n'''
	if primefactorization == None:		#initiate the prime factorization
		p = primes2.pf(n, primelist)
	else:
		p = primefactorization
	product = 1	#Good start for products
	for x in p:
		product*= (x[1]+1) #standard number-theoretic formula for this
	return product
예제 #3
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def phi(n, primelist = None, primefactorization = None):
	'''The Euler phi function, or the quantity
of numbers <= n that are coprime to n.'''
	if primefactorization == None:	#initiate prime factorization
		p = primes2.pf(n, primelist)
	else:
		p = primefactorization
	for x in p:
		n=(n/x[0])*(x[0]-1)	#standard number-theoretic method for computing the Euler phi function
	return n
예제 #4
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def sigma(n, primelist = None, primefactorization = None):
	'''the number-theoretic method for computing the sum of the factors of n'''
	if n == 0:
		print 'infinity' #don't get too clever
	else:
		product = 1	#Good start for products
		if primefactorization == None:	#initiate prime factorization
			p = primes2.pf(n, primelist)
		else:
			p = primefactorization
		for x in p:
			product *= ((x[0]**(x[1]+1))-1)/(x[0]-1)	#standard number-theoretic method for the sigma function
		return product
예제 #5
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def mu(n, primelist = None, primefactorization = None):
	'''mu inversion function'''
	if primefactorization == None:
		p = primes2.pf(n, primelist)
	else:
		p = primefactorization
	t = 1
	for x in p:
		if x[1] == 1:
			t*=(-1)
		else:
			t = 0 
			break
	return t
예제 #6
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def isprimab(n, primelist = None, primefactorization = None):
	'''test whether n is primitive abundant'''
	if n == 0:
		return False
	else:
		if primefactorization == None:		#initiate prime factorization
			p = primes2.pf(n, primelist)
		else:
			p = primefactorization
		m = number_theory2.nu(n, None, p)	#we need to see if n is at all a valid candidate
		if m == 2:
			return True			#all perfect numbers are primitive abundant
		elif m > 2:				#abundant case more complex
			f = impfact(n, None, p)		#find the necessary factors
			x = True			#initiate bool results
			for i in range(len(f)):		#check each factor
				if x:			#if the last factor was deficient . . .
					x = number_theory2.isdeficient(f[i], primelist)	#check if this one is
				else:			#if the last one already wasn't abundant . . .
					break		#no need to check further
			return x 			#return our bool result
		else:
			return False			#clearly, no deficient numbers are primitive abundant