- Vampire number
In
mathematics , a vampire number (or true vampire number) is a compositenatural number "v", with an even number of digits "n", that can be factored into twointegers "x" and "y" each with "n"/2 digits and not both with trailing zeroes, where "v" contains all the digits from "x" and from "y", in any order. "x" and "y" are called the fangs.For example: 1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260. However, 126000 (which can be expressed as 210 × 600) is not, as both 210 and 600 have trailing zeroes.
Vampire numbers first appeared in a 1994 post by
Clifford A. Pickover to theUsenet group sci.math, and the article he later wrote was published in chapter 30 of his book "Keys to Infinity".The vampire numbers are:
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, ...
Multiple fang pairs
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
Variants
Pseudovampire numbers are similar to vampire numbers, except that the fangs of an "n"-digit pseudovampire number need not be of length "n"/2 digits. Pseudovampire numbers can have an odd number of digits, for example 126 = 6×21.
More generally, you can allow more than two fangs. In this case, vampire numbers are numbers "n" which can be factorized using the digits of "n". For example, 1395 = 5×9×31. This sequence starts OEIS|id=A020342::126, 153, 688, 1206, 1255, 1260, 1395, ...
A prime vampire number, as defined by Carlos Rivera in
2002 , is a true vampire number whose fangs are its prime factors. The first few prime vampire numbers are::117067, 124483, 146137, 371893, 536539As of 2006 the largest known is the square (94892254795×1045418+1)2, found by Jens K. Andersen in 2002.References
* Pickover, Clifford A. (1995). "Keys to Infinity". Wiley. ISBN 0-471-19334-8
* [http://groups.google.com/group/sci.math/msg/f17b2281a4aa16da?lr=&ie=UTF-8 Pickover's original post describing vampire numbers]
* Andersen, Jens K. [http://hjem.get2net.dk/jka/math/vampires/ "Vampire Numbers"]
* Rivera, Carlos. [http://www.primepuzzles.net/puzzles/puzz_199.htm "The Prime-Vampire numbers"]External links
*
* Schneider, Walter. [http://web.archive.org/web/20060504235612/http://wschnei.de/digit-related-numbers/vampire.html "Vampire Numbers"]
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