# 33 (number)

33 (thirty-three) is the natural number following 32 and preceding 34.

## In mathematics

33 is the 21st composite number, and 8th distinct semiprime (third of the form $3\times q$ where $q$ is a higher prime). It is one of two numbers to have an aliquot sum of 15 = 3 × 5 — the other being the square of 4 — and part of the aliquot sequence 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, 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. 33 is also the first non-trivial dodecagonal number (like 369, and 561) and the first non-unitary centered dodecahedral number.
It is also the sum of the first four positive factorials, and the sum of the sums of the divisors of the first six positive integers; respectively: ${\begin{aligned}33&=1!+2!+3!+4!=1+2+6+24\\33&=1+3+4+7+6+12\\\end{aligned}}$
It is the first member of the first cluster of three semiprimes 33, 34, 35; the next such cluster is 85, 86, 87. It is also the smallest integer 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 polygons that are used to tile the plane uniformly (the triangle, square, hexagon, octagon, and dodecagon); 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 (1440$_9$), hexadecimal (441$_{16}$) and unotrigesimal (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 and binary (100001).
33 was the second to last number less than 100 whose representation as a sum of three cubes was found (in 2019): $33=8866128975287528^{3}+(-8778405442862239)^{3}+(-2736111468807040)^{3}.$
33 is the sum of the only three locations $n$ in the set of integers $\{1,2,3,...,n\}\in \mathbb {N} ^{+}$ where the ratio of primes to composite numbers is one-to-one (up to $n$) — at, 9, 11, and 13; the latter two represent the fifth and sixth prime numbers, with $9=3^{2}$ the fourth composite. On the other hand, the ratio of prime numbers to non-primes at 33 in the sequence of natural numbers $\mathbb {N} ^{+}$ is ${\tfrac {1}{2}}$, 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 below it (eleven), the product $11\times 33=363$ is the seventh numerator of harmonic number $H_{7}$, 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 of imaginary parts of zeros of the Riemann zeta function, that is also its nearest integer, from an approximate value of $32.93506\ldots$
Written in base-ten, the decimal expansion in the approximation for pi, $\pi \approx 3.141592\ldots$, 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: $\{1,3,5,7,11,15,\mathbf {33} \}.$

## In religion and mythology

* Islamic prayer beads are generally arranged in sets of 33, corresponding to the widespread use of this number in dhikr 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.
* 33 is a master number in New Age numerology, along with 11 and 22.