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== Mathematics == |
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== Mathematics == |
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=== Prime factorization === |
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'''1234''' is a discrete ] with distinct ]s, ] and ].<ref name=A001358>{{Cite OEIS|A001358}}</ref>{{efn|1=It is the 363rd indexed semiprime,<ref name=A001358 /> or the 352nd semiprime that is '''discrete'''.<ref>{{Cite OEIS|A006881}}</ref><br /> }} |
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* '''1234''' is a discrete ] with distinct ]s, ] and ].<ref name="A001358">{{Cite OEIS|A001358}}</ref> |
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=== Decimal properties === |
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* 1234 is the smallest ] that contains the digits ] through ] in ]. |
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==== Concatenation of digits ==== |
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1234 is the smallest ] that contains the digits ] through ] in ].{{efn|1=Ordered by ], the digits "1234" form the fifteenth element in subsets of the ]s.<ref>{{Cite OEIS|A048794}}</ref> }}{{efn|1=1234 has a prime ] of ],<ref name="A002808">{{Cite OEIS|A002808}}</ref> also the index of the fifth ] (of just 1s) in base-ten, <math>R_{1031}</math> (following <math>R_{2}</math>, <math>R_{19}</math>, <math>R_{23}</math>, and <math>R_{317}</math>).<ref>{{Cite OEIS|A004023}}</ref> }} |
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The sum of the base-ten digits of 1234 forms the fourth ] (]). 1234 is more specifically the fourth member of the "Triangle of the gods" ], obtained by concatenating decimal representations of ]s.<ref>{{cite OEIS|A007908|Triangle of the gods}}</ref><ref>{{Cite book|title=A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality|first=Clifford A.|last=Pickover|author-link=Clifford Pickover|publisher=Turner Publishing Company|year=2011|isbn=9781118046074|pages=}}</ref><ref>{{Cite book |author-last=Guy |author-first=Richard K. |author-link=Richard Guy |title=Unsolved Problems in Number Theory |edition=3rd |publisher=Springer |year=2004 |page= |isbn=978-0-387-20860-2 }}</ref> It is also the fifth member of a related integer sequence, obtained from the ] <math>a(n)=10a(n-1)+n</math> starting from <math>a(0)=0</math> and <math>a(1)=1</math>;<ref>{{Cite OEIS|A014824}}</ref> both this sequence and the aforementioned sequence begin in the same way, yet they diverge around their tenth positions. |
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Because it is not divisible by 4, 1234 is the first number in these sequences that is not divisible by its final digit.<ref>{{cite book|page=|title=Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More|first=Matt|last=Parker|author-link=Matt Parker|publisher=Macmillan|year=2014|isbn=9780374275655}}</ref> |
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=== Integer partitions === |
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1234 is the number of integer partitions of ] without all distinct multiplicities,<ref>{{Cite OEIS|A336866}}</ref> as well as the number of partitions of 24 into parts that are prime or ].<ref>{{Cite OEIS|A101049}}</ref> 1234 is the number of "colored" integer partitions of ] such that four colors are used and parts differ by size, or by color.<ref>{{Cite OEIS |A327382}}</ref> It is the number of partitions of ]<sup>2</sup> {{=}} 1089 into exactly four prime numbers.<ref>{{Cite OEIS |A243940}}</ref> |
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Regarding the fourth non-zero decimal ] ], 1234 is its number of strict ] containing the sum of some subset of the parts (as a variation of, sum-full strict partitions),<ref>{{Cite OEIS|A364272}}</ref> as well as the number of partitions of 44 into parts with an ] number of ]s (counted with ]).<ref>{{Cite OEIS|A286218}}</ref> |
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=== Binary strings === |
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1234 is the number of "straight" ] strings of length ] (i.e., the simplest way of representing quantities with binary numbers), equivalently the number of finite ]s of length 22.<ref>{{Cite OEIS|A005598}}</ref> |
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=== Geometric properties === |
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1234 is the number of integer-sided non-degenerate ]s with ] of ].<ref>{{Cite OEIS|A057886}}</ref> |
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==== T-toothpick sequence ==== |
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{{multiple image|perrow = 2|total_width=300 |
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| image1 = T-toothpick after 32 steps (A160172).jpg |
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| image2 = T-toothpick after 33 steps (A160172).jpg |
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| image3 = T-toothpick after 49 steps (A160172).jpg |
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| image4 = T-toothpick after 50 steps (A160172).jpg |
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|footer = T-] (with three equal-sized line segments in T shape) after 32, 33 steps (top), and 49, 50 steps (bottom); respectively. Blue toothpicks represent toothpicks added at that step. |
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}} |
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In a variation of the traditional ], "T-toothpicks" can be formed with three segments of equal length joined at "pivot points" in the shape of a T, which leaves three "endpoints"; these toothpicks are then attached to each other at pivot points with exposed endpoints only (where allowed, see ] for further details).<ref>{{Cite journal |author1-last=Applegate |author1-first=David |author1-link=David Applegate |author2-last=Pol |author2-first=Omar E. |author3-last=Sloane |author3-first=N. J. A. |author3-link=Neil Sloane |title=The toothpick sequence and other sequences from cellular automata. |url=https://oeis.org/A000695/a000695_1.pdf |journal=Congressus Numerantium |volume=206 |publisher=Combinatorial Press |location=Manitoba |year=2010 |pages=183,184 |via=] |mr=2762248 |zbl=1262.11046 |s2cid=12655689 }}</ref><ref name="toothpick">{{Cite OEIS |A160172}}</ref> A square ]-like structure in this sequence is generated at steps (5, 9, 17, 33, ...) while another fractal structure with four squares intersecting a larger square at its corners is generated at steps (6, 12, 25, 49, ...).<ref>{{Cite web |author=David Applegate |editor=N. J. A. Sloane |editor-link=Neil Sloane |title=Explorations of A139250 (Omar Pol's toothpick sequence) and other toothpick-like sequences. |url=https://oeis.org/A139250/a139250.anim.html |website=The ] |publisher=OEIS Foundation }}</ref> At the thirty-second step, the number of toothpicks is 1234, while at the fiftieth step, the number of toothpicks is 2468, or twice 1234.<ref name="toothpick" /> These represent steps that are one step less than an appearing fractal pattern, and one more (respectively; see image). |
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==== Vertex sets ==== |
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There are exactly 1234 ]s in a 4 × 4 ].<ref name=A006506>{{Cite OEIS|A006506}}</ref> This is equivalent with the ways of choosing a subset of positions in a 4 × 4 grid so that no two chosen positions are adjacent horizontally or vertically. For the corresponding problem in one dimension instead of two (choosing points from a sequence with no two adjacent), the number of solutions represents a ].<ref name=A006506 /> |
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== Notes == |
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{{Notelist}} |
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{{Notelist}} |
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