33 (number)

"number 33" redirects here. For other uses, see 33 (disambiguation).

← 32 33 34 →
← 30 31 32 33 34 35 36 37 38 39 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	thirty-three
Ordinal	33rd (thirty-third)
Factorization	3 × 11
Divisors	1, 3, 11, 33
Greek numeral	ΛΓ´
Roman numeral	XXXIII, xxxiii
Binary	1000012
Ternary	10203
Senary	536
Octal	418
Duodecimal	2912
Hexadecimal	2116

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 religion and mythology

• Islamic prayer beads are generally arranged in sets of 33, corresponding to the widespread use of this number in dhikr rituals. Such beads may number 33 in total or three distinct sets of 33 for a total of 99, corresponding to the names of God.
• 33 is a master number in New Age numerology, along with 11 and 22. ^[24]

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.). "SequenceA001748". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA047701(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.). "SequenceA000217(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.). "SequenceA002997(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.). "SequenceA051624(12-gonal (or dodecagonal) number.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-24.
6. Sloane, N. J. A. (ed.). "SequenceA005904(Centered dodecahedral numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-12.
7. Sloane, N. J. A. (ed.). "SequenceA007489(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.). "SequenceA024916(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.). "SequenceA056809". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
10. Sloane, N. J. A. (ed.). "SequenceA005238(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.). "SequenceA005470(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.). "SequenceA057809(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.). "SequenceA001008(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.). "SequenceA00040(The prime numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-12.
16. Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-12.
17. Sloane, N. J. A. (ed.). "SequenceA092783(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.). "SequenceA002410(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.). "SequenceA000396(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.). "SequenceA014976(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.). "SequenceA116582(Numbers from Bhargava's 33 theorem.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-10-09.
24. Sharp, Damian (2001). Simple Numerology: A Simple Wisdom book (A Simple Wisdom Book series). Red Wheel. p. 7. ISBN   978-1573245609.

External links

• Prime Curios! 33 from the Prime Pages
• "33 is the sum of three cubes". Wolfram Alpha .