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Vampire Number

A number $v=xy$ with an Even number $n$ of Digits formed by multiplying a pair of $n/2$-Digit numbers (where the Digits are taken from the original number in any order) $x$ and $y$ together. Pairs of trailing zeros are not allowed. If $v$ is a vampire number, then $x$ and $y$ are called its ``fangs.'' Examples of vampire numbers include

$\displaystyle 1260$ $\textstyle =$ $\displaystyle 21\times 60$  
$\displaystyle 1395$ $\textstyle =$ $\displaystyle 15\times 93$  
$\displaystyle 1435$ $\textstyle =$ $\displaystyle 35\times 41$  
$\displaystyle 1530$ $\textstyle =$ $\displaystyle 30\times 51$  
$\displaystyle 1827$ $\textstyle =$ $\displaystyle 21\times 87$  
$\displaystyle 2187$ $\textstyle =$ $\displaystyle 27\times 81$  
$\displaystyle 6880$ $\textstyle =$ $\displaystyle 80\times 86$  

(Sloane's A014575). There are seven 4-digit vampires, 148 6-digit vampires, and 3228 8-digit vampires. General formulas can be constructed for special classes of vampires, such as the fangs
$\displaystyle x$ $\textstyle =$ $\displaystyle 25\cdot 10^k+1$  
$\displaystyle y$ $\textstyle =$ $\displaystyle 100(10^{k+1}+52)/25,$  

giving the vampire
$\displaystyle v$ $\textstyle =$ $\displaystyle xy=(10^{k+1}+52)10^{k+2}+100(10^{k+1}+52)/25$  
  $\textstyle =$ $\displaystyle x^* \cdot 10^{k+2}+t$  
  $\textstyle =$ $\displaystyle 8(26+5\cdot 10^k)(1+25\cdot 10^k),$  

where $x^*$ denotes $x$ with the Digits reversed (Roushe and Rogers).

Pickover (1995) also defines pseudovampire numbers, in which the multiplicands have different numbers of digits.


Pickover, C. A. ``Vampire Numbers.'' Ch. 30 in Keys to Infinity. New York: Wiley, pp. 227-231, 1995.

Pickover, C. A. ``Vampire Numbers.'' Theta 9, 11-13, Spring 1995.

Pickover, C. A. ``Interview with a Number.'' Discover 16, 136, June 1995.

Roushe, F. W. and Rogers, D. G. ``Tame Vampires.'' Undated manuscript.

Sloane, N. J. A. Sequence A014575 in ``The On-Line Version of the Encyclopedia of Integer Sequences.''

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© 1996-9 Eric W. Weisstein