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

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[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.