Misplaced Pages

1234 (number): Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editNext edit →Content deleted Content addedVisualWikitext
Revision as of 20:05, 11 August 2024 editXOR'easter (talk | contribs)Extended confirmed users32,777 edits Undid revision 1239791869 by Radlrb (talk): the discussion being pointed to as a reason to include this trivia did not actually discuss it; keeping an article at AfD is not an argument to preserve it in that state like a cryogenically frozen headTag: Undo← Previous edit Revision as of 20:15, 11 August 2024 edit undoRadlrb (talk | contribs)Extended confirmed users9,249 edits You are right, I used the wrong wording. I meant, many of the properties listed here that were kept after the discussion to keep this article are points whose admission on a number article are actively being discussed, and therefore should not be used as criteria yet (for/against inclusion). Also, two other people returned the information by reverting two other editors that attempted to remove it; clearly there is no consensus to delete, so please stop trying to delete this, which joins in th...Tags: Reverted Mobile edit Mobile web editNext edit →
Line 9: Line 9:


== Mathematics == == Mathematics ==
=== Prime factorization ===

* '''1234''' is a discrete ] with distinct ]s, ] and ].<ref name="A001358">{{Cite OEIS|A001358}}</ref> '''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 /> }}
* '''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 /> }}
=== Decimal properties ===
* 1234 is the smallest ] that contains the digits ] through ] in ]. * 1234 is the smallest ] that contains the digits ] through ] in ].
==== Concatenation of digits ====
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> }}
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.
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>
=== Integer partitions ===
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>
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>
=== Binary strings ===
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>
=== Geometric properties ===
1234 is the number of integer-sided non-degenerate ]s with ] of ].<ref>{{Cite OEIS|A057886}}</ref>
==== T-toothpick sequence ====
{{multiple image|perrow = 2|total_width=300
| image1 = T-toothpick after 32 steps (A160172).jpg
| image2 = T-toothpick after 33 steps (A160172).jpg
| image3 = T-toothpick after 49 steps (A160172).jpg
| image4 = T-toothpick after 50 steps (A160172).jpg
|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.
}}
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).
==== Vertex sets ====
There are exactly 1234 ]s in a 4 &times; 4 ].<ref name=A006506>{{Cite OEIS|A006506}}</ref> This is equivalent with the ways of choosing a subset of positions in a 4 &times; 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 />
== Notes ==
{{Notelist}} {{Notelist}}


Revision as of 20:15, 11 August 2024

This article is about the number 1234. For the year, see 1234. Natural number
← 1233 1234 1235 →
← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinalone thousand two hundred thirty-four
Ordinal1234th
(one thousand two hundred thirty-fourth)
Factorization2 × 617
Greek numeral,ΑΣΛΔ´
Roman numeralMCCXXXIV, mccxxxiv
Binary100110100102
Ternary12002013
Senary54146
Octal23228
Duodecimal86A12
Hexadecimal4D216

1234 is the natural number following 1233, and preceding 1235.

A 2012 study of frequently-used personal identification numbers (PIN) found that, among 4-digit pin codes, 1234 is the most frequently chosen.

Mathematics

Prime factorization

1234 is a discrete semiprime with distinct prime factors, 2 and 617.

Decimal properties

Concatenation of digits

1234 is the smallest whole number that contains the digits 1 through 4 in decimal. The sum of the base-ten digits of 1234 forms the fourth triangular number (10). 1234 is more specifically the fourth member of the "Triangle of the gods" sequence, obtained by concatenating decimal representations of positive integers. It is also the fifth member of a related integer sequence, obtained from the recurrence relation a ( n ) = 10 a ( n − 1 ) + n {\displaystyle a(n)=10a(n-1)+n} starting from a ( 0 ) = 0 {\displaystyle a(0)=0} and a ( 1 ) = 1 {\displaystyle a(1)=1} ; both this sequence and the aforementioned sequence begin in the same way, yet they diverge around their tenth positions. Because it is not divisible by 4, 1234 is the first number in these sequences that is not divisible by its final digit.

Integer partitions

1234 is the number of integer partitions of 24 without all distinct multiplicities, as well as the number of partitions of 24 into parts that are prime or semiprime. 1234 is the number of "colored" integer partitions of 12 such that four colors are used and parts differ by size, or by color. It is the number of partitions of 33 = 1089 into exactly four prime numbers. Regarding the fourth non-zero decimal repdigit 44, 1234 is its number of strict partitions containing the sum of some subset of the parts (as a variation of, sum-full strict partitions), as well as the number of partitions of 44 into parts with an odd number of prime divisors (counted with multiplicity).

Binary strings

1234 is the number of "straight" binary strings of length 22 (i.e., the simplest way of representing quantities with binary numbers), equivalently the number of finite Sturmian words of length 22.

Geometric properties

1234 is the number of integer-sided non-degenerate quadrilaterals with perimeter of 50.

T-toothpick sequence

T-Toothpick sequence (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.

In a variation of the traditional Toothpick sequence, "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 A160172 for further details). A square fractal-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, ...). 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. These represent steps that are one step less than an appearing fractal pattern, and one more (respectively; see image).

Vertex sets

There are exactly 1234 independent vertex sets in a 4 × 4 square grid. 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 Fibonacci number.

Notes

  1. It is the 363rd indexed semiprime, or the 352nd semiprime that is discrete.
  2. It is the 363rd indexed semiprime, or the 352nd semiprime that is discrete.
  3. Ordered by standard statistical (or Yates) order, the digits "1234" form the fifteenth element in subsets of the natural numbers.
  4. 1234 has a prime composite index of 1031, also the index of the fifth prime repunit (of just 1s) in base-ten, R 1031 {\displaystyle R_{1031}} (following R 2 {\displaystyle R_{2}} , R 19 {\displaystyle R_{19}} , R 23 {\displaystyle R_{23}} , and R 317 {\displaystyle R_{317}} ).

References

  1. Berry, N. (September 3, 2012). "PIN analysis". Data Genetics. As cited by Nisbet, Alastair; Kim, Maria (December 2016). "An analysis of chosen alarm code pin numbers & their weakness against a modified brute force attack". In Johnstone, M. (ed.). The Proceedings of 14th Australian Information Security Management Conference. Perth, Australia: Edith Cowan University. pp. 21–29. doi:10.4225/75/58a69fd2a8b03.{{cite conference}}: CS1 maint: date and year (link)
  2. ^ Sloane, N. J. A. (ed.). "Sequence A001358". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. Sloane, N. J. A. (ed.). "Sequence A006881". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. Sloane, N. J. A. (ed.). "Sequence A006881". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. Sloane, N. J. A. (ed.). "Sequence A048794". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. Sloane, N. J. A. (ed.). "Sequence A002808". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. Sloane, N. J. A. (ed.). "Sequence A004023". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. Sloane, N. J. A. (ed.). "Sequence A007908 (Triangle of the gods)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. Pickover, Clifford A. (2011). A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality. Turner Publishing Company. pp. 10–11. ISBN 9781118046074.
  10. Guy, Richard K. (2004). Unsolved Problems in Number Theory (3rd ed.). Springer. p. 15. ISBN 978-0-387-20860-2.
  11. Sloane, N. J. A. (ed.). "Sequence A014824". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  12. Parker, Matt (2014). 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. Macmillan. p. 8. ISBN 9780374275655.
  13. Sloane, N. J. A. (ed.). "Sequence A336866". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  14. Sloane, N. J. A. (ed.). "Sequence A101049". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  15. Sloane, N. J. A. (ed.). "Sequence A327382". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  16. Sloane, N. J. A. (ed.). "Sequence A243940". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  17. Sloane, N. J. A. (ed.). "Sequence A364272". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  18. Sloane, N. J. A. (ed.). "Sequence A286218". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  19. Sloane, N. J. A. (ed.). "Sequence A005598". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  20. Sloane, N. J. A. (ed.). "Sequence A057886". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  21. Applegate, David; Pol, Omar E.; Sloane, N. J. A. (2010). "The toothpick sequence and other sequences from cellular automata" (PDF). Congressus Numerantium. 206. Manitoba: Combinatorial Press: 183, 184. MR 2762248. S2CID 12655689. Zbl 1262.11046 – via OEIS.
  22. ^ Sloane, N. J. A. (ed.). "Sequence A160172". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  23. David Applegate. N. J. A. Sloane (ed.). "Explorations of A139250 (Omar Pol's toothpick sequence) and other toothpick-like sequences". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  24. ^ Sloane, N. J. A. (ed.). "Sequence A006506". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Integers
−1
0s
  •  0 
  •  1 
  •  2 
  •  3 
  •  4 
  •  5 
  •  6 
  •  7 
  •  8 
  •  9 
100s
200s
300s
400s
500s
600s
700s
800s
900s
1000s
Category: