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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 ${\displaystyle a(n)=10a(n-1)+n}$ starting from ${\displaystyle a(0)=0}$ and ${\displaystyle a(1)=1}$; 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

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. Ordered by standard statistical (or Yates) order, the digits "1234" form the fifteenth element in subsets of the natural numbers.
3. 1234 has a prime composite index of 1031, also the index of the fifth prime repunit (of just 1s) in base-ten, ${\displaystyle R_{1031}}$ (following ${\displaystyle R_{2}}$, ${\displaystyle R_{19}}$, ${\displaystyle R_{23}}$, and ${\displaystyle R_{317}}$).

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 A048794". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. Sloane, N. J. A. (ed.). "Sequence A002808". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. Sloane, N. J. A. (ed.). "Sequence A004023". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. Sloane, N. J. A. (ed.). "Sequence A007908 (Triangle of the gods)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. 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.
9. Guy, Richard K. (2004). Unsolved Problems in Number Theory (3rd ed.). Springer. p. 15. ISBN 978-0-387-20860-2.
10. Sloane, N. J. A. (ed.). "Sequence A014824". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
11. 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.
12. Sloane, N. J. A. (ed.). "Sequence A336866". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
13. Sloane, N. J. A. (ed.). "Sequence A101049". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
14. Sloane, N. J. A. (ed.). "Sequence A327382". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
15. Sloane, N. J. A. (ed.). "Sequence A243940". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
16. Sloane, N. J. A. (ed.). "Sequence A364272". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
17. Sloane, N. J. A. (ed.). "Sequence A286218". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
18. Sloane, N. J. A. (ed.). "Sequence A005598". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
19. Sloane, N. J. A. (ed.). "Sequence A057886". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
20. 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.
21. ^ Sloane, N. J. A. (ed.). "Sequence A160172". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
22. 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.
23. ^ Sloane, N. J. A. (ed.). "Sequence A006506". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

• Integers