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vampire numbers
A number  $n$  with  $2k$  digits is called vampire there exists two numbers  $x$  and  $y$  of  $k$  digits each such that  $n=x\cdot y$  and that  $x$  and  $y$  together have the same digits of  $n$.

Moreover, to exclude numbers which can be trivially obtained from smaller ones,  $x$  and  $y$  cannot end both in '0'.

The first vampire numbers are 1260 = 2160, 1395 = 1593, 1435 = 3541, 1530 = 3051 and 1827 = 2187.

It can be shown easily that when divided by 9 a vampire number has a remainder equal to 0 or 4.

Since  $n$  is a vampire number, the numbers  $x$  and  $y$  are called fangs.

There are many vampire numbers which have more than one pair of fangs, for example 125460 = 246510 = 204615 and
13078260 = 16208073 = 18637020 = 20706318.

J. K. Andersen has found, among many other results, a 70-digit vampire number with 100025 different fang pairs.

The vampire numbers were introduced by Clifford A. Pickover in 1994. Note that other similar definition are possible (and sometimes used). For example, it is possible to relax the constraint on the number of digits of  $x$  and  $y$, or have more terms in the product, like in  $1395 = 5\cdot9\cdot31$.

The first vampire numbers are 1260, 1395, 1435, 1530, 1827, 2187, 6880, 102510, 104260, 105210, 105264, 105750, 108135, 110758 more terms

Vampire numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 2187 378450 10396800 + 4968449928 aban 1206000297 1255000198 1255000950 + 8199000297 abundant 1260 1530 6880 + 49867236 Achilles 10396800 10716300 11926728 + 5894137611 admirable 229648 alternating 2187 105210 329656 + 58703472 amenable 1260 6880 104260 + 96977920 apocalyptic 1435 1827 2187 arithmetic 1395 1435 1827 + 841995 binomial 14340690 1465813440 1563429820 + 7589181600 brilliant 117067 124483 146137 + 79168819 c.decagonal 1209712351 c.heptagonal 3074749861 c.octagonal 19847025 51480625 1116695889 1480325625 c.triangular 7036628359 congruent 6880 102510 105264 + 939658 cube 26198073 Cunningham 16265088 1052158968 1328675400 + 6769833840 Curzon 1530 105210 105750 + 95438970 cyclic 117067 124483 146137 + 536539 d-powerful 193257 263074 284598 729688 de Polignac 124483 371893 489955 + 83119297 decagonal 12526290 1223547930 1319814405 + 4305885970 deficient 1395 1435 1827 + 939658 dig.balanced 136948 145314 152685 + 89598460 Duffinian 2187 117067 124483 + 536539 economical 1435 2187 108135 + 19847025 emirpimes 124483 146137 371893 + 77329939 equidigital 1435 110758 117067 + 19554277 eRAP 8509256892 evil 1260 1530 1827 + 96977920 Friedman 1260 1395 1435 + 939658 frugal 2187 108135 126846 + 75296875 gapful 1260 1395 1530 + 9953948970 happy 102510 105210 105264 + 829696 Harshad 1260 1530 102510 + 9953948970 heptagonal 260338 1025024629 1643485540 + 7349168574 hexagonal 14340690 1820789685 2510959545 highly composite 1260 hoax 117067 124483 136525 + 94765918 iban 174370 304717 inconsummate 102510 110758 135828 + 841995 interprime 1827 108135 116725 + 89901900 Jordan-Polya 186624 junction 102510 105210 116725 + 94619952 Lehmer 81992911 1508692531 4869114151 + 5763104941 lucky 1395 1435 1827 + 792585 Lynch-Bell 1395 384912 modest 1827 180297 14300271 + 1976443902 nonagonal 2189713059 nude 1395 163944 186624 + 61612992 octagonal 35164480 1299376408 1475812840 2970642136 odious 1395 1435 2187 + 96399072 pandigital 15249780 80993700 81739588 + 8210953476 pentagonal 1128017682 1627054870 1710399852 + 5258884176 pernicious 1395 1435 2187 + 815958 persistent 1023657984 1023679584 1043568297 + 8143697520 power 2187 186624 19847025 + 5267275776 powerful 2187 186624 10396800 + 5894137611 practical 1260 1530 6880 + 829696 prim.abundant 229648 10323225 11786350 + 45185872 pronic 1260 1178926560 1287553806 + 3482593182 Proth 180225 pseudoperfect 1260 1530 6880 + 841995 Rhonda 13985257 Ruth-Aaron 16672216 21109374 32533816 + 3906853470 Saint-Exupery 4580802720 self 104260 132430 135837 + 86139495 semiprime 117067 124483 146137 + 79168819 Smith 6880 117067 124483 + 94765918 sphenic 1435 110758 131242 + 83119297 square 186624 19847025 21068100 + 5267275776 super Niven 15003000 1050021000 1500030000 super-d 105210 105264 105750 + 809919 superabundant 1260 tau 1260 120600 156240 + 96399072 taxicab 1456273728 1702564920 tetrahedral 1563429820 triangular 14340690 1465813440 1658332845 + 7589181600 Ulam 126027 135837 146952 + 362992 unprimeable 1260 102510 104260 + 939658 untouchable 6880 104260 105210 + 815958 upside-down 23428678 71673493 wasteful 1260 1395 1530 + 939658 Zuckerman 186624 11632896 1411323264 + 4139413632 Zumkeller 1260 1530 6880 zygodrome 44995500 2266335544 5522776600