Inrecreational mathematics, avampire number (ortrue vampire number) is acompositenatural number with an even number ofdigits, that can be factored into two natural numbers each with half as many digits as the original number, where the two factors contain precisely all the digits of the original number, in any order, counting multiplicity. The two factors cannot both have trailing zeroes. The first vampire number is 1260 = 21 × 60.^[1][2]

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 theconcatenation 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 apermutation of the${\displaystyle 2k}$ digits of${\displaystyle N}$. The two numbers${\displaystyle A}$ and${\displaystyle B}$ are called thefangs of${\displaystyle N}$.

Vampire numbers were first described in a 1994 post byClifford A. Pickover to theUsenet group sci.math,^[3] and the article he later wrote was published in chapter 30 of his bookKeys 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, ... (sequenceA014575 in theOEIS)

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

Vampire numbers also exist for bases other than base 10. For example, a vampire number inbase 12 is 10392BA45768 = 105628 × BA3974, where A means ten and B means eleven. Another example in the same base is a vampire number with three fangs, 572164B9A830 = 8752 × 9346 × A0B1. An example with four 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 fromthe original on 2017-10-14. Retrieved2013-04-08.

• Base-dependent integer sequences
• Metaphors referring to monsters

• Articles with short description
• Short description is different from Wikidata