Vampire Numbers

 

Vampire Numbers:

A Vampire number is a composite natural number with 2n digits, that can be factored into two natural numbers each with n digits, where the two factors together contain precisely all the digits of the original number, in any order. The two factors cannot both have trailing zeroes. For example, 1260 is a Vampire number as

1260 = 21 60

where factors 21 and 60 are cal or a pair of fangs.

Although 126000 = 210 600 where the two factors together contain precisely all the digits of the original number, is not a Vampire number because both factors 210 and 600 have trailing zeroes.

Vampire numbers were first described by Clifford A. Pickover [1]

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  A014575).

All the four-digit Vampires are given below:

1260 = 21 60

1395 = 15 93

1435 = 35 41

1530 = 30 51

1827 = 21 27

2187 = 27 81

6880 = 80 86

Many sequences of infinite Vampire numbers can be constructed by placing zeros at appropriate places, such as:

1260 = 21 60, 120600 = 201 600, 12006000 = 2001 6000, ...

1530 = 30 51, 150300 = 300 501, 15003000 = 3000 5001, ... and so on.

It may be noted that Vampire numbers are little different than Cab numbers [2] in some respect because of the way in which these numbers are defined. However, there is an overlap, as certain Cab numbers are also Vampire numbers also.
For example, all 2n - digit Cab numbers of the type AB CD, ABC DEF and ABCD EFGH are all Vampire numbers also.

A Vampire number can have multiple distinct pairs of fangs.
For example, Vampire numbers having two distinct pairs of fangs [OEIS A048936] are:

125460 = 204 615 = 246 510

11930170 = 1301 9170 = 1310 9107

12054060 = 2004 6015 = 2406 5010

12417993 = 1317 9429 = 1347 9219

Examples of Vampire numbers having three distinct pairs of fangs [OEIS A048937] are:

13078260 = 1620 8073 = 1863 7020 = 2070 6318

107650322640 = 153204 702660 = 140532 766020 = 200760 536214

113024597400 = 125100 903474 = 152100 743094 = 257400 439101

119634515208 = 195351 612408 = 234156 510918 = 285513 419016

Examples of Vampire numbers having four and five distinct pairs of fangs are:

16758243290880

= 1982736 8452080

= 2123856 7890480

= 2751840 6089832

= 2817360 5948208

and

24959017348650

= 2947050 8469153

= 2949705 8461530

= 4125870 6049395

= 4129587 6043950

= 4230765 5899410

Double Vampire Numbers:

A double Vampire number is a Vampire number whose fangs are also Vampire numbers. The smallest double Vampire number is

1047527295416280 = 25198740 41570622 = (2940 8571) (5601 7422)

Note that 25198740 and 41570622 are also Vampire numbers as

25198740 = 2940 8571

41570622 = 5601 7422

Pseudovampires :

Pickover (1995) also defines pseudovampire, in which the multiplicands have different numbers of digits. So, pseudovampire can have an odd number of digits, for example,

126 = 6 21

In addition, pseudovampire numbers can have more than two fang. For example,

1395 = 5 9 31

The sequence of pseudovampire numbers is:

126, 153, 688, 1206, 1255, 1260, 1395, 1435, 1503, 1530, 1827, 2187, 3159, 3784, 6880, 10251, 10255, 10426, 10521, 10525, 10575, 11259, 11439, 11844, 11848, 12006, 12060, 12384, 12505, 12546, 12550, 12595, 12600, 12762, 12768, ... (OEIS  A020342).

Palindromic Vampire Numbers:

A palindromic vampire number is a vampire number which is palindromic also. Some examples of palindromic Vampire numbers are:

410591195014 = 451154 910091

597168861795 = 756681 789195

27923155132972 = 2932571 9521732

28573944937582 = 3252593 8784974

29160744706192 = 4709072 6192461

61548755784516 = 7555548 8146167

76399855899367 = 7966739 9589853

1081919999191801 = 11899811 90919091

1340875115780431 = 15114407 88715033

1409195995919041 = 15499451 90919091

1869372002739681 = 22806393 81967017

2071929889291702 = 22788722 90919091

2327608118067232 = 26723312 87100286

2824301991034282 = 30421241 92839802

2914336886334192 = 31868322 91449336

3061939779391603 = 33677633 90919091

It is interesting to note the palindromic Vampire numbers which are multiples of 90919091. On dividing such palindromic Vampire numbers by 90919091, another palindromic number is obtained.
In other words, one fang of these palindromic Vampire numbers is also palindromic.

Square Vampire Numbers:

A Vampire number which is also a perfect square can be termed as square Vampire number. For example, 186624 is a square Vampire number as,

4322 = 186624 = 216 864

It may be noted that for a square Vampire number, its fangs need not be equal i.e., fangs may not be square root of square Vampire number.

The square Vampire number 5267275776 is the square of its fangs i.e.,

725762 = 5267275776 = 72576 72576

Some other examples of square Vampire numbers are:

44552 = 19847025 = 2475 8019

44552 = 21068100 = 2601 8100

44552 = 51480625 = 6125 8405

44552 = 1116695889 = 19881 56169

44552 = 1119571600 = 11711 95600

44552 = 1315367824 = 18134 72536

44552 = 1480325625 = 23085 64125

44552 = 2363709924 = 24309 97236

44552 = 103789642896 = 107388 966492

44552 = 114842365456 = 134456 854126

44552 = 117974388676 = 171737 686948

44552 = 135829102500 = 152100 893025

44552 = 137926818225 = 222831 618975

44552 = 162889731216 = 197136 826281

44552 = 165252006144 = 406512 406512

44552 = 172455817284 = 415278 415278

44552 = 202905202500 = 220500 920205

44552 = 203716822500 = 260100 783225

44552 = 213166890000 = 263169 810000

Prime Vampire Numbers:

A prime Vampire number as defined by Carlos Rivera [3] is a Vampire number whose fangs are prime numbers. For example, 117067 is a prime Vampire number as,

117067 = 167 701

Where 167 and 701 are prime numbers.

The sequence of prime Vampire numbers is:

117067, 124483, 146137, 371893, 536539, 10349527, 10429753, 10687513, 11722657, 11823997, 12451927, 12484057, 12894547, 13042849, 14145799, 14823463, 17204359, 18517351, 18524749, 18647023, 19262587, ... (OEIS  A289911).

Interestingly, there are no prime Vampire numbers with three fangs. The smallest examples of 3-digit, k-fang prime Vampire numbers for k = 4, 5 and 6 are given below:

The smallest 3-digit, 4-fang prime Vampire number is:

103997964977 = 379 409 677 991

The smallest 3-digit, 5-fang prime Vampire number is:

100688737751983 = 307 607 719 853 881

The smallest 3-digit, 6-fang prime Vampire number is:

100967417475216533 = 503 641 653 701 727 941

We can also have prime Vampire numbers which are perfect squares also. For example,

495915232 = 2459319153459529 = 49591523 49591523 Where 49591523 is a prime number.

Some other examples of prime Vampire numbers which are perfect squares also are given are:

2512099504480801 = 50120849 50120849 = 501208492

3395758728913321 = 58273139 58273139 = 582731392

3893627338729969 = 62398937 62398937 = 623989372

5129508768706921 = 71620589 71620589 = 716205892

8186379410403769 = 90478613 90478613 = 904786132

170147428389340249 = 412489307 412489307 = 4124893072

189598345243224241 = 435428921 435428921 = 4354289212

271971550510512889 = 521508917 521508917 = 5215089172

334573968623758249 = 578423693 578423693 = 5784236932

571691675535320209 = 756102953 756102953 = 7561029532

---------------------------------------------------------------------------

References:

[1] Pickover, C. A. Keys to Infinity. New York: Wiley, p. 227-231, 1995.

[2] Gupta, S. S. Cab Numbers."

[3] Rivera, C. Problems & Puzzles: Puzzle 199-The Prime-Vampire Numbers."

[4] Sloane, N. J. A. OEIS A014575, A020342, A048936, A048937, and A289911 in The On-Line Encyclopedia of Integer Sequences.



In Association with Amazon.com

This page is created on 12 July, 2023.

Back Home