1234 (number)

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

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

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

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 $$a(n)=10a(n-1)+n$$ starting from $$a(0)=0$$ and $$a(1)=1$$; 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.

Square root
Though not strictly a schizophrenic number in base ten, the square root of 1234


 * $$\sqrt {1234} \approx 35.128336140500 \ldots$$

is the first in its sequence to have a composite integer part, of 35. The sequence of integer parts of schizophrenic numbers (with odd index) is self-similar, featuring repunits. In like fashion, all even terms in this sequence starting with the square root of 1234 follow the pattern "35, 351, 3513, ..."

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 332 = 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 &times; 4 square grid. 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 Fibonacci number.