33 (thirty-three) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following
32 and preceding
34.
In mathematics
33 is the 21st
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...
, and 8th distinct
semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers.
Because there are infinitely many prime n ...
(third of the form
where
is a higher prime). It is one of two numbers to have an
aliquot sum
In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself.
That is,
s(n)=\sum_ d \, .
It can be used to characterize the prime numbers, perfect numbers, sociabl ...
of
15 = 3 × 5 — the other being the
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
of
4 — and part of the
aliquot sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
Def ...
of
9 = 3
2 in the aliquot tree (33,
15, 9,
4,
3,
2,
1).
It is the largest positive integer that cannot be expressed as a sum of different
triangular numbers
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
, and it is the largest of twelve integers that are not the sum of five non-zero squares; on the other hand, the 33rd triangular number
561 is the first
Carmichael number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
: b^n\equiv b\pmod
for all integers . The relation may also be expressed in the form:
: b^\equiv 1\pmod
for all integers b ...
. 33 is also the first non-trivial
dodecagonal number
In mathematics, a dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for ''n'' is given by the formula
:D_=5n^2 - 4n
The first few dodecagonal numbers are:
: 0, 1, 12, 33, 64, 105, 156, 217, 288, ...
(like 369, and 561) and the first non-unitary
centered dodecahedral number.
It is also the sum of the first four positive
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
s, and the sum of the sums of the divisors of the first six
positive integers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
; respectively:
It is the first member of the first cluster of three
semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers.
Because there are infinitely many prime n ...
s 33,
34,
35; the next such cluster is
85,
86,
87. It is also the smallest
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
such that it and the next two integers all have the same number of divisors (four).
33 is the number of
unlabeled planar simple graphs with five
nodes.
There are only five
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s that are used to tile the
plane uniformly (the
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
,
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
,
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
,
octagon
In geometry, an octagon () is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...
, and
dodecagon
In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.
Regular dodecagon
A regular polygon, regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry ...
); the total number of sides in these is: 3 + 4 + 6 + 8 + 12 = 33.
33 is equal to the sum of the squares of the digits of its own square in
nonary
A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information.
Although ''ternary'' ...
(1440
9),
hexadecimal
Hexadecimal (also known as base-16 or simply hex) is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbo ...
(441
16) and untrigesimal (144
31). For numbers greater than
1, this is a rare property to have in more than one
base. It is also a palindrome in both
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
and
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two values (0 and 1) for each digit
* Binary function, a function that takes two arguments
* Binary operation, a mathematical op ...
(100001).
33 was the second to last number less than
100 whose representation as a
sum of three cubes was found (in 2019):
33 is the sum of the only three locations
in the set of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s
where the ratio of primes to
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...
s is one-to-one (up to
) — at,
9,
11, and
13; the latter two represent the fifth and sixth prime numbers, with
the fourth composite. On the other hand, the ratio of prime numbers to non-primes at 33 in the sequence of
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s
is
, where there are (inclusively)
11 prime numbers and
22 non-primes (i.e., when including
1).
Where 33 is the seventh number divisible by the number of
prime numbers
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
below it (eleven), the product
is the seventh numerator of
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
, where specifically, the previous such numerators are
49 and
137, which are respectively the thirty-third
composite and prime numbers.
33 is the fifth
ceiling
A ceiling is an overhead interior roof that covers the upper limits of a room. It is not generally considered a structural element, but a finished surface concealing the underside of the roof structure or the floor of a story above. Ceilings can ...
of
imaginary parts of
zeros of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
, that is also its
nearest integer, from an
approximate
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...
value of
Written in
base-ten, the
decimal expansion
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\cdots b_0.a_1a_2\cdots
Here is the decimal separator ...
in the approximation for
pi,
, has
0 as its 33rd digit, the first such single-digit string.
A positive
definite quadratic integer matrix represents all
odd numbers when it contains ''at least'' the set of seven integers:
In religion and mythology
* Islamic
prayer beads are generally arranged in sets of 33, corresponding to the widespread use of this number in
dhikr
(; ; ) is a form of Islamic worship in which phrases or prayers are repeatedly recited for the purpose of remembering God. It plays a central role in Sufism, and each Sufi order typically adopts a specific ''dhikr'', accompanied by specific ...
rituals. Such beads may number 33 in total or three distinct sets of 33 for a total of 99, corresponding to the
names of God
There are various names of God, many of which enumerate the various Quality (philosophy), qualities of a Supreme Being. The English word ''God (word), god'' (and its equivalent in other languages) is used by multiple religions as a noun to ref ...
.
* 33 is a master number in
New Age
New Age is a range of Spirituality, spiritual or Religion, religious practices and beliefs that rapidly grew in Western world, Western society during the early 1970s. Its highly eclecticism, eclectic and unsystematic structure makes a precise d ...
numerology
Numerology (known before the 20th century as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, ...
, along with 11 and 22.
Notes
References
External links
Prime Curios! 33from the
Prime Pages
The PrimePages is a website about prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is ...
*
{{DEFAULTSORT:33 (Number)
Integers