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In number theory, a vampire number (or true vampire number) is a composite natural number with an even number of digits, that can be factored into two natural numbers each with half as many digits as the original number, where the two factors contain precisely all the digits of the original number, in any order, counting multiplicity. The two factors cannot both have trailing zeroes. The first vampire number is 1260 = 21 × 60. Definition Let N {\displaystyle N} be a natural number with 2 k {\displaystyle 2k} digits: N = n 2 k n 2 k − 1 . . . n 1 {\displaystyle N={n_{2k}}{n_{2k-1}}...{n_{1}}} Then N {\displaystyle N} is a vampire number if and only if there exist two natural numbers A {\displaystyle A} and B {\displaystyle B} , each with k {\displaystyle k} digits: A = a k a k − 1 . . . a 1 {\displaystyle A={a_{k}}{a_{k-1}}...{a_{1}}} B = b k b k − 1 . . . b 1 {\displaystyle B={b_{k}}{b_{k-1}}...{b_{1}}} such that A × B = N {\displaystyle A\times B=N} , a 1 {\displaystyle a_{1}} and b 1 {\displaystyle b_{1}} are not both zero, and the 2 k {\displaystyle 2k} digits of the concatenation of A {\displaystyle A} and B {\displaystyle B} ( a k a k − 1 . . . a 2 a 1 b k b k − 1 . . . b 2 b 1 ) {\displaystyle ({a_{k}}{a_{k-1}}...{a_{2}}{a_{1}}{b_{k}}{b_{k-1}}...{b_{2}}{b_{1}})} are a permutation of the 2 k {\displaystyle 2k} digits of N {\displaystyle N} . The two numbers A {\displaystyle A} and B {\displaystyle B} are called the fangs of N {\displaystyle N} . Vampire numbers were first described in a 1994 post by Clifford A. Pickover to the Usenet group sci.math, and the article he later wrote was published in chapter 30 of his book Keys to Infinity. Examples 1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260 and the digits of the concatenation of the two factors (2160) are a permutation of the digits of the original number (1260). However, 126000 (which can be expressed as 21 × 6000 or 210 × 600) is not a vampire number, since although 126000 = 21 × 6000 and the digits (216000) are a permutation of the original number, the two factors 21 and 6000 do not have the correct number of digits. Furthermore, although 126000 = 210 × 600, both factors 210 and 600 have trailing zeroes. The first few vampire numbers are: 1260 = 21 × 60 1395 = 15 × 93 1435 = 35 × 41 1530 = 30 × 51 1827 = 21 × 87 2187 = 27 × 81 6880 = 80 × 86 102510 = 201 × 510 104260 = 260 × 401 105210 = 210 × 501The 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, ... (sequence A014575 in the OEIS)There are many known sequences of infinitely many vampire numbers following a pattern, such as: 1530 = 30 × 51, 150300 = 300 × 501, 15003000 = 3000 × 5001, ...Al Sweigart calculated all the vampire numbers that have at most 10 digits. Multiple fang pairs A vampire number can have multiple distinct pairs of fangs. The first of infinitely many vampire numbers with 2 pairs of fangs: 125460 = 204 × 615 = 246 × 510The first with 3 pairs of fangs: 13078260 = 1620 × 8073 = 1863 × 7020 = 2070 × 6318The first with 4 pairs of fangs: 16758243290880 = 1982736 × 8452080 = 2123856 × 7890480 = 2751840 × 6089832 = 2817360 × 5948208The first with 5 pairs of fangs: 24959017348650 = 2947050 × 8469153 = 2949705 × 8461530 = 4125870 × 6049395 = 4129587 × 6043950 = 4230765 × 5899410Variants Pseudovampire numbers (disfigurate vampire numbers) are similar to vampire numbers, except that the fangs of an n-digit pseudovampire number need not be of length n/2 digits. Pseudovampire numbers can have an odd number of digits, for example 126 = 6 × 21. More generally, more than two fangs are allowed. In this case, vampire numbers are numbers n which can be factorized using the digits of n. For example, 1395 = 5 × 9 × 31. This sequence starts (sequence A020342 in the OEIS): 126, 153, 688, 1206, 1255, 1260, 1395, ...A vampire prim.... Discover the Al Sweigart popular books. Find the top 100 most popular Al Sweigart books.

Best Seller Al Sweigart Books of June 2022

Routineaufgaben mit Python automatisieren book summary, reviews and downlod

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The Recursive Book of Recursion

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Automate the Boring Stuff with Python, 2nd Edition

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Wielka księga małych projektów w Pythonie. 81 łatwych praktycznych programów book summary, reviews and downlod

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Best Seller Al Sweigart Audio Books of June 2022

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