# 33 (number)

 ← 32 33 34 →
Cardinalthirty-three
Ordinal33rd
(thirty-third)
Factorization3 × 11
Divisors1, 3, 11, 33
Greek numeralΛΓ´
Roman numeralXXXIII
Binary1000012
Ternary10203
Senary536
Octal418
Duodecimal2912

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 ${\displaystyle 3\times q}$ where ${\displaystyle q}$ 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] {\displaystyle {\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.[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] ${\displaystyle 33=8866128975287528^{3}+(-8778405442862239)^{3}+(-2736111468807040)^{3}.}$

33 is the sum of the only three locations ${\displaystyle n}$ in the set of integers ${\displaystyle \{1,2,3,...,n\}\in \mathbb {N} ^{+}}$ where the ratio of primes to composite numbers is one-to-one (up to ${\displaystyle n}$) — at, 9, 11, and 13; the latter two represent the fifth and sixth prime numbers, with ${\displaystyle 9=3^{2}}$ the fourth composite. On the other hand, the ratio of prime numbers to non-primes at 33 in the sequence of natural numbers ${\displaystyle \mathbb {N} ^{+}}$ is ${\displaystyle {\tfrac {1}{2}}}$, 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),[13] the product ${\displaystyle 11\times 33=363}$ is the seventh numerator of harmonic number ${\displaystyle H_{7}}$,[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 ${\displaystyle 32.93506\ldots }$[17][18][19][a]

Written in base-ten, the decimal expansion in the approximation for pi, ${\displaystyle \pi \approx 3.141592\ldots }$, has 0 as its 33rd digit, the first such single-digit string.[21][b]

A positive definite quadratic integer matrix represents all odd numbers when it contains at least the set of seven integers: ${\displaystyle \{1,3,5,7,11,15,\mathbf {33} \}.}$[22][23]

## In science

### Astronomy

• Messier object M33, a magnitude 7.0 galaxy in the constellation Triangulum, also known as the Triangulum Galaxy.
• The New General Catalogue object NGC 33, a double star in the constellation Pisces
• The smallest dwarf planet in the Solar System is Ceres, which is also the 33rd largest celestial body in the Solar System, comprising about one-third of the mass in the asteroid belt.[24]
• 33 is the number of years that it takes for the Lunar phase to return to its original position in relation to the Solar calendar. A Lunar month (Synodic) contains 29.53 days. A twelve-month lunar year contains 354.36 days.[25] A solar year (Tropical year) totals 365.24 days.[26] The lunar year is therefore 10.88 days shorter than the 12-month solar year. As each year passes, the lunar month trails 10.88 days behind the solar year. On the turn of the 33rd year, the lunar month is approximately 359.04 days, close to one whole year behind the solar calendar from the original position measured, thus it has a 33-year cycle in relation to the solar year. Where the lunar year ${\displaystyle 354.36=l}$ and the solar year ${\displaystyle 365.24=s}$, then ${\displaystyle s-l=d}$ and ${\displaystyle 33l=33(s-d)}$. Many cultures and civilisations have based their calendar on the lunar cycles including the Athenian Attic calendar[27] and the Islamic Calendar, the Hijri calendar based on lunar observation.[28]

## In technology

• In reference to gramophone records, 33 refers to a type of record by its revolution speed of 33+13 revolutions per minute. 33s are also known as long playing records, or LPs. See: 78 and 45
• The ITU country code for the French telephone numbering plan area

## In media

• The number 33 is featured in Dark, a German science fiction television series following intertwined storylines over increments of 33 years.
• The 33 is a biographical disaster film based on the real events of a mining disaster that occurred in 2010, where a group of 33 miners became trapped inside the San José Mine in Chile.[35]
• 33 is the first episode of the re-imagined military science fiction television series Battlestar Galactica. The fleet are forced to execute a faster-than-light (FTL) jump every 33 minutes to evade the Cylons.

## In other fields

Thirty-three is:

• The number printed on all Rolling Rock beer labels.
• Pabst Blue Ribbon Beer used to be advertised as "Blended 33 to 1".
• The name brand of a mass-market lager beer, "33" Export, brewed and distributed in West Africa.
• The namesake of the private club, Club 33, located in Disneyland's New Orleans Square.
• The number of workers trapped, all of whom were rescued, during the 2010 Copiapó mining accident.
• The 33 Orientales were a group of Uruguay's national Independence Heroes that liberate the country in 1825 from the Brazilian Empire, the name is due for the leaders all 33 Degree Masons (The Thirty-Three Orientals), one of Uruguay's national states and its capital city is named "Treinta y Tres" after them
• The modern Russian alphabet consists of 33 letters.[36]
• Georgian is presently written in a 33-letter alphabet.[37]
• The number of bogatyrs who emerged from the sea in the Russian fairy-tale Tsar Saltan.
• The number of digits required to uniquely specify every human currently alive in binary code.

## Notes

1. ^ These first seven digits in this approximation end in 6 and generate a sum of 28 (the seventh triangular number), numbers which represent the first and second perfect numbers, respectively (where-also, the sum between these two numbers is 34, with 35 = 7 + 28).[20]
2. ^ Where 3 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.

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A001748". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Sloane, N. J. A. (ed.). "Sequence A047701 (All positive numbers that are not the sum of 5 nonzero squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-09.
3. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers: a(n) is the binomial(n+1,2) equal to n*(n+1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-15.
4. ^ Sloane, N. J. A. (ed.). "Sequence A002997 (Carmichael numbers: composite numbers n such that a^(n-1) congruent 1 (mod n) for every a coprime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-15.
5. ^ Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-24.
6. ^ Sloane, N. J. A. (ed.). "Sequence A005904 (Centered dodecahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-12.
7. ^ Sloane, N. J. A. (ed.). "Sequence A007489 (a(n) is Sum_{k equal to 1..n} k!.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-12.
8. ^ Sloane, N. J. A. (ed.). "Sequence A024916 (a(n) is Sum_{k equal to 1..n} k*floor(n/k); also Sum_{k equal to 1..n} sigma(k) where sigma(n) is the sum of divisors of n (A000203).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-12.
9. ^ Sloane, N. J. A. (ed.). "Sequence A056809". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
10. ^ Sloane, N. J. A. (ed.). "Sequence A005238 (Numbers k such that k, k+1 and k+2 have the same number of divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
11. ^ Sloane, N. J. A. (ed.). "Sequence A005470 (Number of unlabeled planar simple graphs with n nodes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-12.
12. ^ Booker, Andrew R. (2019). "Cracking the problem with 33". arXiv:1903.04284 [math.NT].
13. ^ Sloane, N. J. A. (ed.). "Sequence A057809 (Numbers n such that pi(n) divides n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-30.
14. ^ Sloane, N. J. A. (ed.). "Sequence A001008 (Numerators of harmonic numbers H(n) as the Sum_{i equal to 1..n} 1/i.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-12.
15. ^ Sloane, N. J. A. (ed.). "Sequence A00040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-12.
16. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-12.
17. ^ Sloane, N. J. A. (ed.). "Sequence A092783 (Ceiling of imaginary parts of zeros of Riemann zeta function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-06-01.
18. ^ Sloane, N. J. A. (ed.). "Sequence A002410 (Nearest integer to imaginary part of n-th zero of Riemann zeta function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-06-02.
19. ^ Odlyzko, Andrew. "The first 100 (non trivial) zeros of the Riemann Zeta function [AT&T Labs]". Andrew Odlyzko: Home Page. UMN CSE. Retrieved 2024-01-16.
20. ^ Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-06-02.
21. ^ Sloane, N. J. A. (ed.). "Sequence A014976 (Successive locations of zeros in decimal expansion of Pi.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-30.
22. ^ Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. Vol. 239 (1st ed.). Springer. pp. 312–314. doi:10.1007/978-0-387-49923-9. ISBN 978-0-387-49922-2. OCLC 493636622. Zbl 1119.11001.
23. ^ Sloane, N. J. A. (ed.). "Sequence A116582 (Numbers from Bhargava's 33 theorem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-09.
24. ^ Williams, Matt (August 24, 2015). "What is the asteroid belt?". Phys.org. Science X. Retrieved 2023-09-22.
25. ^ http://adsabs.harvard.edu/full/1992JHAS...23...32S The Length of the Lunar Month, Schaefer, B. E.
26. ^ http://adsabs.harvard.edu/full/1991JRASC..85..121B The Tropical Year and Solar Calendar, Borkowski, K. M.
27. ^ worldhistory.org The Athenian Calendar
28. ^ https://eclipse.gsfc.nasa.gov/SEhelp/calendars.html Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann
29. ^ Insights #517, October 8, 2010.
30. ^ de Vries, Ad (1976). Dictionary of Symbols and Imagery. Amsterdam: North-Holland Publishing Company. pp. 462. ISBN 978-0-7204-8021-4.
31. ^ Ghazzālī; Karim, Fazlul (1978). "Imam Gazzali's Ihya Ulum-id-din: pt. 1 and 2. The book of constructive virtues". Sind Sagar Academy. Retrieved 21 March 2018 – via Google Books.
32. ^ Sharp, Damian (2001). Simple Numerology: A Simple Wisdom book (A Simple Wisdom Book series). Red Wheel. p. 7. ISBN 978-1573245609.
33. ^ "Dedicated umpire stayed at the plate for 32 innings. - Free Online Library". www.thefreelibrary.com. Retrieved 2020-08-21.
34. ^ Cary, Tim (2015-02-14). "10 of the Longest Winning Streaks in Sports History". Sportscasting | Pure Sports. Retrieved 2020-08-21.
35. ^ "THE 33 | British Board of Film Classification". www.bbfc.co.uk. Retrieved 2020-08-21.
36. ^ "Russian Language Alphabet - listen online and practice pronunciation". Russian Step By Step Books Natasha Alexandrova. Retrieved 2020-08-21.
37. ^ "Georgian Alphabet | Georgian Language, Alphabet and Pronunciation". www.ocf.berkeley.edu. Retrieved 2020-08-21.