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Natural number From Wikipedia, the free encyclopedia
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 where is a higher prime).[1] 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 = 32 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;[2] on the other hand, the 33rd triangular number 561 is the first Carmichael number.[3][4] 33 is also the first non-trivial dodecagonal number (like 369, and 561)[5] and the first non-unitary centered dodecahedral number.[6]
It is also the sum of the first four positive factorials,[7] and the sum of the sum of the divisors of the first six positive integers; respectively:[8]
It is the first member of the first cluster of three semiprimes 33, 34, 35; the next such cluster is 85, 86, 87.[9] It is also the smallest integer such that it and the next two integers all have the same number of divisors (four).[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. 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):[12]
33 is the sum of the only three locations in the set of integers 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 the fourth composite. On the other hand, the ratio of prime numbers to non-primes at 33 in the sequence of natural numbers 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 below it (eleven),[13] the product 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][lower-alpha 1]
Written in base-ten, the decimal expansion in the approximation for pi, , has 0 as its 33rd digit, the first such single-digit string.[21][lower-alpha 2]
A positive definite quadratic integer matrix represents all odd numbers when it contains at least the set of seven integers: [22][23]
Thirty-three is:
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