Vampire number

Let ${\displaystyle N}$ be a natural number with ${\displaystyle 2k}$ digits:

${\displaystyle N={n_{2k}}{n_{2k-1}}...{n_{1}}}$

Then ${\displaystyle N}$ is a vampire number if and only if there exist two natural numbers ${\displaystyle A}$ and ${\displaystyle B}$, each with ${\displaystyle k}$ digits:

${\displaystyle A={a_{k}}{a_{k-1}}...{a_{1}}}$

${\displaystyle B={b_{k}}{b_{k-1}}...{b_{1}}}$

such that ${\displaystyle A\times B=N}$, ${\displaystyle a_{1}}$ and ${\displaystyle b_{1}}$ are not both zero, and the ${\displaystyle 2k}$ digits of the concatenation of ${\displaystyle A}$ and ${\displaystyle B}$ ${\displaystyle ({a_{k}}{a_{k-1}}...{a_{2}}{a_{1}}{b_{k}}{b_{k-1}}...{b_{2}}{b_{1}})}$ are a permutation of the ${\displaystyle 2k}$ digits of ${\displaystyle N}$. The two numbers ${\displaystyle A}$ and ${\displaystyle B}$ are called the fangs of ${\displaystyle N}$.

Vampire numbers were first described in a 1994 post by Clifford A. Pickover to the Usenet group sci.math,[3] and the article he later wrote was published in chapter 30 of his book Keys to Infinity.[4]

1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260 and the digits of the concatenation of the two factors (2160) are a permutation of the digits of the original number (1260).

However, 126000 (which can be expressed as 21 × 6000 or 210 × 600) is not a vampire number, since although 126000 = 21 × 6000 and the digits (216000) are a permutation of the original number, the two factors 21 and 6000 do not have the correct number of digits. Furthermore, although 126000 = 210 × 600, both factors 210 and 600 have trailing zeroes.

The first few vampire numbers are:

1260 = 21 × 60

1395 = 15 × 93

1435 = 35 × 41

1530 = 30 × 51

1827 = 21 × 87

2187 = 27 × 81

6880 = 80 × 86

102510 = 201 × 510

104260 = 260 × 401

105210 = 210 × 501

The sequence of vampire numbers is:

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, ...

Al Sweigart calculated all the vampire numbers that have at most 10 digits.[5]

A vampire number can have multiple distinct pairs of fangs. The first of infinitely many vampire numbers with 2 pairs of fangs:

125460 = 204 × 615 = 246 × 510

The first with 3 pairs of fangs:

13078260 = 1620 × 8073 = 1863 × 7020 = 2070 × 6318

The first with 4 pairs of fangs:

16758243290880 = 1982736 × 8452080 = 2123856 × 7890480 = 2751840 × 6089832 = 2817360 × 5948208

The first with 5 pairs of fangs:

24959017348650 = 2947050 × 8469153 = 2949705 × 8461530 = 4125870 × 6049395 = 4129587 × 6043950 = 4230765 × 5899410

Pseudovampire numbers (disfigurate vampire numbers) are similar to vampire numbers, except that the fangs of an n-digit pseudovampire number need not be of length n/2 digits. Pseudovampire numbers can have an odd number of digits, for example 126 = 6 × 21.

More generally, more than two fangs are allowed. In this case, vampire numbers are numbers n which can be factorized using the digits of n. For example, 1395 = 5 × 9 × 31. This sequence starts (sequence A020342 in the OEIS):

126, 153, 688, 1206, 1255, 1260, 1395, ...

A vampire prime or prime vampire number, as defined by Carlos Rivera in 2002,[6] is a true vampire number whose fangs are its prime factors. The first few vampire primes are:

117067, 124483, 146137, 371893, 536539

As of 2007 the largest known is the square (94892254795 × 10¹⁰³²⁹⁴ + 1)², found by Jens K. Andersen in September, 2007.[2]

A double vampire number is a vampire number which has fangs that are also vampire numbers, an example of such a number is 1047527295416280 = 25198740 × 41570622 = (2940 × 8571) × (5601 × 7422) which is the smallest double vampire number.

A Roman numeral vampire number is vampire number that uses Roman numerals instead of base-10. An example of this number is II × IV = VIII.

Vampire numbers also exist for bases other than base 10.

For example, a vampire number in base 12 is 10392BA45768 = 105628 × BA3974, where A mens ten and B means eleven. Another example in the same base is a vampire number with 3 fangs, 572164B9A830 = 8752 × 9346 × A0B1. One example with 4 fangs is 3715A6B89420 = 763 × 824 × 905 × B1A. In these examples, all 12 digits are used exactly once.

• Friedman number

• Sweigart, Al. Vampire Numbers Visualized
• Grime, James; Copeland, Ed. "Vampire numbers". Numberphile. Brady Haran. Archived from the original on 2017-10-14. Retrieved 2013-04-08.