Vampire Number

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:

  1. Has a pair number of digits. Lets call the number of digits : n
  2. You can obtain the number by multiplying two integers, x and y, each with n/2 digits. x and y are the fangs.
  3. Both fangs cannot end simultaneously in 0.
  4. 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:

  1. Rossetta Code
  2. Wikipedia – Vampire Number
  3. Stackoverflow
  4. Python Docs on itertools

This article is contributed by Subhajit Saha. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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