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 program ` `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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