vampire number

Let $N$ be a natural number with $2k$ digits:

:$N\; =\; \{n\_\{2k\}\}\{n\_\{2k-1\}\}...\{n\_1\}$

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

:$A\; =\; \{a\_k\}\{a\_\{k-1\}\}...\{a\_1\}$

:$B\; =\; \{b\_k\}\{b\_\{k-1\}\}...\{b\_1\}$

such that $A\; \backslash times\; B\; =\; N$, $a\_1$ and $b\_1$ are not both zero, and the $2k$ digits of the concatenation of $A$ and $B$ $(\{a\_k\}\{a\_\{k-1\}\}...\{a\_2\}\{a\_1\}\{b\_k\}\{b\_\{k-1\}\}...\{b\_2\}\{b\_1\})$ are a permutation of the $2k$ digits of $N$. The two numbers $A$ and $B$ are called the fangs of $N$.

Vampire numbers were first described in a 1994 post by Clifford A. Pickover to the Usenet group sci.math,[http://groups.google.com/group/sci.math/msg/f17b2281a4aa16da?lr=&ie=UTF-8 Pickover's original post describing vampire numbers] and the article he later wrote was published in chapter 30 of his book Keys to Infinity.{{cite book |last1=Pickover |first1=Clifford A. |title=Keys to Infinity |date=1995 |publisher=Wiley |isbn=0-471-19334-8}}

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, ... {{OEIS|id=A014575}}

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.{{cite web |last1=Sweigart |first1=Al |title=Vampire Numbers Visualized |url=http://coffeeghost.net/2011/07/19/vampire-numbers-visualized/}}

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 in base 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

{{reflist}}

• Sweigart, Al. [http://coffeeghost.net/2011/07/19/vampire-numbers-visualized/ Vampire Numbers Visualized]
• {{cite web|last1=Grime|first1=James|title=Vampire numbers|url=http://www.numberphile.com/videos/vampire_numbers.html|last2=Copeland|first2=Ed|work=Numberphile|publisher=Brady Haran|access-date=2013-04-08|archive-date=2017-10-14|archive-url=https://web.archive.org/web/20171014053632/http://www.numberphile.com/videos/vampire_numbers.html|url-status=dead}}

{{Classes of natural numbers}}

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Category:Base-dependent integer sequences