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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

1260=21·60
(1)
1395=15·93
(2)
1435=35·41
(3)
1530=30·51
(4)
1827=21·87
(5)
2187=27·81
(6)
6880=80·86
(7)

(OEIS A014575). The 8-digit vampire numbers are 10025010, 10042510, 10052010, 10052064, 10081260, ... (OEIS A048938) and the 10-digit vampire numbers are 1000174288, 1000191991, 1000198206, 1000250010, ... (OEIS A048939). The numbers of 2n-digit vampires are 0, 7, 148, 3228, ... (OEIS A048935).

Vampire numbers having two distinct pairs of fangs include

125460=204·615=246·510
(8)
11930170=1301·9170=1310·9107
(9)
12054060=2004·6015=2406·5010
(10)

(OEIS A048936).

Vampire numbers having three distinct pairs of fangs include

 13078260=1620·8073=1863·7020=2070·6318.
(11)

(OEIS A048937).

The first vampire numbers with four pairs of fangs are

16758243290880=1982736·8452080
(12)
=2123856·7890480
(13)
=2751840·6089832
(14)
=2817360·5948208
(15)

and

18762456533040=2558061·7334640
(16)
=3261060·5753484
(17)
=3587166·5230440
(18)
=3637260·5158404,
(19)

and the first vampire number with five pairs of fangs is

24959017348650=2947050·8469153
(20)
=2949705·8461530
(21)
=4125870·6049395
(22)
=4129587·6043950
(23)
=4230765·5899410
(24)

(J. K. Andersen, pers. comm., May 4, 2003).

General formulas can be constructed for special classes of vampires, such as the fangs

x=25·10^k+1
(25)
y=100(10^(k+1)+52)/25,
(26)

giving the vampire

v=xy
(27)
=(10^(k+1)+52)10^(k+2)+100(10^(k+1)+52)/25
(28)
=x^*·10^(k+2)+y
(29)
=8(26+5·10^k)(1+25·10^k),
(30)

where x^* denotes x with the digits reversed (Roush and Rogers 1997-1998).

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


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References

Anderson, J. K. "Vampire Numbers." http://hjem.get2net.dk/jka/math/vampires/.Childs, J. "Vampire Numbers!" http://www.grenvillecc.ca/faculty/jchilds/vampire.htm.Childs, J. "Vampire Numbers! Part 2." http://www.grenvillecc.ca/faculty/jchilds/vampire2.htm.Childs, J. "Vampire Numbers--Information Summary--Part 3." http://www.grenvillecc.ca/faculty/jchilds/vampire3.htm.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.Rivera, C. "Problems & Puzzles: Puzzle 199-The Prime-Vampire Numbers." http://www.primepuzzles.net/puzzles/puzz_199.htm.Roush, F.W.; Rogers, D. G. "Tame Vampires." Math. Spectrum 30, 37-39, 1997-1998.Update a linkSchneider, W. "Vampire Numbers." http://wschnei.de/digit-related-numbers/vampire.html Sloane, N. J. A. Sequences A014575, A048933, A048934, A048935, A048936, A048937, A048938, and A048939 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Vampire Number

Cite this as:

Weisstein, Eric W. "Vampire Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VampireNumber.html

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