33 (thirty-three) is the natural number following 32 and preceding 34.
33 is the 21st composite number, and 8th distinct semiprime (third of the form
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;[2] on the other hand, the 33rd triangular number 561 is the first Carmichael number.
[10] 33 is the number of unlabeled planar simple graphs with five nodes.
[11] 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 (14409), hexadecimal (44116) and unotrigesimal (14431).
For numbers greater than 1, this is a rare property to have in more than one base.
33 was the second to last number less than 100 whose representation as a sum of three cubes was found (in 2019):[12]
where the ratio of primes to composite numbers is one-to-one (up to
) — at, 9, 11, and 13; the latter two represent the fifth and sixth prime numbers, with
On the other hand, the ratio of prime numbers to non-primes at 33 in the sequence of natural numbers
, where there are (inclusively) 11 prime numbers and 22 non-primes (i.e., when including 1).
is the seventh numerator of harmonic number
,[14] where specifically, the previous such numerators are 49 and 137, which are respectively the thirty-third composite and prime numbers.
[15][16] 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
[17][18][19][a] Written in base-ten, the decimal expansion in the approximation for pi,
[21][b] A positive definite quadratic integer matrix represents all odd numbers when it contains at least the set of seven integers: