Vampire Number

A number with an Even number of Digits formed by multiplying a pair of -Digit numbers (where the Digits are taken from the original number in any order) and together. Pairs of trailing zeros are not allowed. If is a vampire number, then and 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

denotes

with the Digits reversed (Roushe and Rogers).

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

References

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.'' http://www.research.att.com/~njas/sequences/eisonline.html.