Introduction to Vampire Number and its implementation using python.

**Introduction**

In mathematics, a vampire number (or true vampire number) is a composite natural number v, with an even number of digits n, that can be factored into two integers x and y each with n/2 digits and not both with trailing zeroes, where v contains precisely all the digits from x and from y, in any order, counting multiplicity. *x and y are called the fangs.* [Source Wiki]

Examples:

- 1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260.
- 126000 (which can be expressed as 21 × 6000 or 210 × 600) is not, as 21 and 6000 do not have the correct length, and both 210 and 600 have trailing zeroes

The vampire numbers are:

1260, 1395, 1435, 1530, 1827, 2187, 6880, 102510, 104260, 105210, 105264, 105750, 108135, 110758, 115672, 116725, 117067, 118440, 120600, 123354, 124483, 125248, 125433, 125460, 125500, … (sequence A014575 in the OEIS)

There are many known sequences of infinitely many vampire numbers following a pattern, such as:

1530 = 30×51, 150300 = 300×501, 15003000 = 3000×5001, …

Condition for a number to be Vampire Number:

- Has a pair number of digits. Lets call the number of digits : n
- You can obtain the number by multiplying two integers, x and y, each with n/2 digits. x and y are the fangs.
- Both fangs cannot end simultaneously in 0.
- The number can be made with all digits from x and y, in any order and only using each digit once.

**Pseudocode**

if digitcount is odd return false if digitcount is 2 return false for A = each permutation of length digitcount/2 selected from all the digits, for B = each permutation of the remaining digits, if either A or B starts with a zero, continue if both A and B end in a zero, continue if A*B == the number, return true

# Python code to check if a number is Vampire # and printing Vampire numbers upto n using # it import itertools as it # function to get the required fangs of the # vampire number def getFangs(num_str): # to get all possible orderings of order that # is equal to the number of digits of the # vampire number num_iter = it.permutations(num_str, len(num_str)) # creating the possible pairs of number by # brute forcing, then checking the condition # if it satisfies what it takes to be the fangs # of a vampire number for num_list in num_iter: v = ''.join(num_list) x, y = v[:int(len(v)/2)], v[int(len(v)/2):] # if numbers have trailing zeroes then skip if x[-1] == '0' and y[-1] == '0': continue # if x * y is equal to the vampire number # then return the numbers as its fangs if int(x) * int(y) == int(num_str): return x,y return False # function to check whether the given number is # vampire or not def isVampire(m_int): # converting the vampire number to string n_str = str(m_int) # if no of digits in the number is odd then # return false if len(n_str) % 2 == 1: return False # getting the fangs of the number fangs = getFangs(n_str) if not fangs: return False return True # main driver programm n = 16000 for test_num in range(n): if isVampire(test_num): print ("{}".format(test_num), end = ", ")

Output:

1260, 1395, 1435, 1530, 1827, 2187, 6880,

Refer to numberphile for more details:

**References:**

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