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

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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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.

This page is created on 12 July, 2023.

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