Vampire Numbers

There is more than one type of vampire number but the first type that I'll deal with in this post belongs to OEIS A014575:

A014575: Vampire numbers (definition 2): numbers \(n\) with an even number of digits which have a factorization \(n = i \times j\) where \(i\) and \(j\) have the same number of digits and the multiset of the digits of \(n\) coincides with the multiset of the digits of \(i\) and \(j\).

Examples are \(1260=21 \times 60\) and \(939658=953 \times 986\). The two relevant divisors of a vampire number are called its fangs and the numbers we are dealing with here have two fangs.

Up to one million, the vampire numbers are (permalink):

1260, 1395, 1435, 1530, 1827, 2187, 6880, 102510, 104260, 105210, 105264, 105750, 108135, 110758, 115672, 116725, 117067, 118440, 120600, 123354, 124483, 125248, 125433, 125460, 125460, 125500, 126027, 126846, 129640, 129775, 131242, 132430, 133245, 134725, 135828, 135837, 136525, 136948, 140350, 145314, 146137, 146952, 150300, 152608, 152685, 153436, 156240, 156289, 156915, 162976, 163944, 172822, 173250, 174370, 175329, 180225, 180297, 182250, 182650, 186624, 190260, 192150, 193257, 193945, 197725, 201852, 205785, 211896, 213466, 215860, 216733, 217638, 218488, 226498, 226872, 229648, 233896, 241564, 245182, 251896, 253750, 254740, 260338, 262984, 263074, 284598, 284760, 286416, 296320, 304717, 312475, 312975, 315594, 315900, 319059, 319536, 326452, 329346, 329656, 336550, 336960, 338296, 341653, 346968, 361989, 362992, 365638, 368550, 369189, 371893, 378400, 378418, 378450, 384912, 386415, 392566, 404968, 414895, 416650, 416988, 428980, 429664, 447916, 456840, 457600, 458640, 475380, 486720, 489159, 489955, 498550, 516879, 529672, 536539, 538650, 559188, 567648, 568750, 629680, 638950, 673920, 679500, 729688, 736695, 738468, 769792, 789250, 789525, 792585, 794088, 809919, 809964, 815958, 829696, 841995, 939658

As can be seen, \(1260\) is the first four digit vampire number and \(6880=80 \times 86\) is the last. The first six digit number is \(102510=201 \times 510\) and the last is \(939658\). Of the 156 numbers listed above, it can be seen that \(125460\) appears twice and this is because it has two representations viz. \(204 \times 615\) and \(246 \times 510\). 

\(13078260\) is an example of a vampire number that has three representations:$$ \begin{align} 13078260 &=1620 \times 8073\\&=1863 \times 7020 \\&=2070 \times 6318 \end{align}$$Wolfram Mathworld has examples of numbers that can be represented in four and five different ways.

Another type of vampire number is listed in OEIS A020342:

A020342: Vampire numbers: (definition 1): \(n\) has a nontrivial factorization using \(n\)'s digits. Nontrivial means that there must be at least two factors.

An example is \(126 = 6 \times 21\) and \(39784=8 \times 4973\). The initial members of this sequence are up to 40000 (permalink):

126, 153, 688, 1206, 1255, 1260, 1260, 1395, 1435, 1503, 1530, 1530, 1827, 2187, 3159, 3784, 6880, 6880, 10251, 10255, 10426, 10521, 10525, 10575, 11259, 11844, 11848, 12006, 12060, 12060, 12384, 12505, 12546, 12550, 12550, 12595, 12600, 12600, 12600, 12762, 12843, 12955, 12964, 13243, 13545, 13950, 13950, 14035, 14350, 14350, 15003, 15030, 15030, 15246, 15300, 15300, 15300, 15435, 15624, 15795, 16272, 17325, 17428, 17437, 17482, 18225, 18265, 18270, 18270, 19026, 19215, 21375, 21586, 21753, 21870, 21870, 25105, 25375, 25474, 25510, 28476, 29632, 31509, 31590, 31590, 33655, 33696, 36855, 37840, 37840, 37845, 39784

There are 712 numbers in the range up to one million, with some repeated numbers of course. All of the members of OEIS A014575 are in this sequence but the conditions (that the number must have an even number of digits and have its two divisors of equal length and not both ending in zero) have been relaxed. Consequently, any number \(n\) in the sequence will also have \(n \times 10\) in the sequence.