33 (number)

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

← 32 33 34 →
Cardinalthirty-three
Ordinal33rd
(thirty-third)
Factorization3 × 11
Divisors1, 3, 11, 33
Greek numeralΛΓ´
Roman numeralXXXIII
Binary1000012
Ternary10203
Senary536
Octal418
Duodecimal2912
Hexadecimal2116

In mathematics

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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][a]

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][b]

A positive definite quadratic integer matrix represents all odd numbers when it contains at least the set of seven integers:  [22][23]

In science

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Astronomy

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In technology

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  • 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 religion and mythology

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In sports

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In media

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  • 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.[31]
  • 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

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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.[32]
  • Georgian is presently written in a 33-letter alphabet.[33]
  • 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.

See also

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Notes

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  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

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  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. ^ Insights #517, October 8, 2010.
  26. ^ de Vries, Ad (1976). Dictionary of Symbols and Imagery. Amsterdam: North-Holland Publishing Company. pp. 462. ISBN 978-0-7204-8021-4.
  27. ^ 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.
  28. ^ Sharp, Damian (2001). Simple Numerology: A Simple Wisdom book (A Simple Wisdom Book series). Red Wheel. p. 7. ISBN 978-1573245609.
  29. ^ "Dedicated umpire stayed at the plate for 32 innings. - Free Online Library". www.thefreelibrary.com. Retrieved 2020-08-21.
  30. ^ Cary, Tim (2015-02-14). "10 of the Longest Winning Streaks in Sports History". Sportscasting | Pure Sports. Retrieved 2020-08-21.
  31. ^ "THE 33 | British Board of Film Classification". www.bbfc.co.uk. Retrieved 2020-08-21.
  32. ^ "Russian Language Alphabet - listen online and practice pronunciation". Russian Step By Step Books Natasha Alexandrova. Retrieved 2020-08-21.
  33. ^ "Georgian Alphabet | Georgian Language, Alphabet and Pronunciation". www.ocf.berkeley.edu. Retrieved 2020-08-21.
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