1234 (number)

This article is about the number 1234. For the year, see 1234.

← 1233 1234 1235 →
List of numbers Integers ← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinal	one thousand two hundred thirty-four
Ordinal	1234th (one thousand two hundred thirty-fourth)
Factorization	2 × 617
Greek numeral	,ΑΣΛΔ´
Roman numeral	MCCXXXIV
Binary	100110100102
Ternary	12002013
Senary	54146
Octal	23228
Duodecimal	86A12
Hexadecimal	4D216

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.

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 $a(n)=10a(n-1)+n$ 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.

Mock rational properties

As a schizophrenic number in base ten, the square root of 1234 displays two-digit repdigits within the first five digits of its fractional part, where there is

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

Among these types of numbers, this expansion contains the earliest appearance of repdigits after the decimal point (that is, before larger terms in the sequence), where it is also the first to have a composite integer part, of 35.

Integer partitions

The number of strict partitions of 44 containing the sum of some subset of the parts is 1234 (as a variation of, sum-full strict partitions).

1234 is also the number of solutions to other integer partitions, including:

• The number of integer partitions of 24 without all distinct multiplicities.
• The number of partitions of 33² = 1089 into exactly four prime numbers.
• The number of partitions of 24 into parts that are prime or semiprime.
• The number of "colored" integer partitions of 12 such that four colors are used and parts differ by size, or by color.

1234 is furthermore 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.