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,
432^{2} = 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.,
72576^{2} = 5267275776 = 72576 × 72576
Some other examples of square Vampire numbers are:
4455^{2} = 19847025 = 2475 × 8019
4455^{2} = 21068100 = 2601 × 8100
4455^{2} = 51480625 = 6125 × 8405
4455^{2} = 1116695889 = 19881 × 56169
4455^{2} = 1119571600 = 11711 × 95600
4455^{2} = 1315367824 = 18134 × 72536
4455^{2} = 1480325625 = 23085 × 64125
4455^{2} = 2363709924 = 24309 × 97236
4455^{2} = 103789642896 = 107388 × 966492
4455^{2} = 114842365456 = 134456 × 854126
4455^{2} = 117974388676 = 171737 × 686948
4455^{2} = 135829102500 = 152100 × 893025
4455^{2} = 137926818225 = 222831 × 618975
4455^{2} = 162889731216 = 197136 × 826281
4455^{2} = 165252006144 = 406512 × 406512
4455^{2} = 172455817284 = 415278 × 415278
4455^{2} = 202905202500 = 220500 × 920205
4455^{2} = 203716822500 = 260100 × 783225
4455^{2} = 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,
49591523^{2} = 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 = 50120849^{2}
3395758728913321 = 58273139 × 58273139 = 58273139^{2}
3893627338729969 = 62398937 × 62398937 = 62398937^{2}
5129508768706921 = 71620589 × 71620589 = 71620589^{2}
8186379410403769 = 90478613 × 90478613 = 90478613^{2}
170147428389340249 = 412489307 × 412489307 = 412489307^{2}
189598345243224241 = 435428921 × 435428921 = 435428921^{2}
271971550510512889 = 521508917 × 521508917 = 521508917^{2}
334573968623758249 = 578423693 × 578423693 = 578423693^{2}
571691675535320209 = 756102953 × 756102953 = 756102953^{2}
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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.
This page is created on 12 July, 2023.