It is the largest positive integer that cannot be expressed as a sum of differenttriangular 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 number561 is the firstCarmichael number.[3][4] 33 is also the first non-trivialdodecagonal number (like 369, and 561)[5] and the first non-unitarycentered dodecahedral number.[6]
It is also the sum of the first four positivefactorials,[7] and the sum of the sums of the divisors of the first sixpositive integers; respectively:[8]
It is the first member of the first cluster of threesemiprimes 33,34,35; the next such cluster is85,86,87.[9] It is also the smallestinteger such that it and the next two integers all have the same number of divisors (four).[10]
33 is equal to the sum of the squares of the digits of its own square innonary (14409),hexadecimal (44116) and untrigesimal (14431). For numbers greater than1, this is a rare property to have in more than onebase. It is also a palindrome in bothdecimal andbinary (100001).
33 was the second to last number less than100 whose representation as asum of three cubes was found (in 2019):[12]
33 is the sum of the only three locations in the set ofintegers where the ratio of primes tocomposite numbers is one-to-one (up to) — at,9,11, and13; 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 ofnatural numbers is, where there are (inclusively)11 prime numbers and22 non-primes (i.e., when including1).
Where 33 is the seventh number divisible by the number ofprime numbers below it (eleven),[13] the product is the seventh numerator ofharmonic number,[14] where specifically, the previous such numerators are49 and137, which are respectively the thirty-thirdcomposite and prime numbers.[15][16]
Islamicprayer beads are generally arranged in sets of 33, corresponding to the widespread use of this number indhikr rituals. Such beads may number 33 in total or three distinct sets of 33 for a total of 99, corresponding to thenames of God.
"33" appears on every bottle ofRolling Rock beer. No definitive meaning has been identified, though theories include that is the year ofProhibition ending in the US and the number of letters in quality statement also printed on the bottle. The company website states 'nobody knows' the reason, but list several popular theories.[25]
^These first seven digits in this approximation end in6 and generate a sum of28 (the seventhtriangular number), numbers which represent the first and secondperfect numbers, respectively (where-also, the sum between these two numbers is34, with35 =7 + 28).[20]
^Where3 is the first digit of pi in decimal representation, the sum between the sixteenth and seventeenth instances (16 + 17 = 33) of a zero-string are at the 165th and 168th digits, positions whose values generate a sum of 333, and difference of 3.
