30 (number)

← 29 30 31 →
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Cardinal	thirty
Ordinal	30th (thirtieth)
Factorization	2 × 3 × 5
Divisors	1, 2, 3, 5, 6, 10, 15, 30
Greek numeral	Λ´
Roman numeral	XXX, xxx
Binary	111102
Ternary	10103
Senary	506
Octal	368
Duodecimal	2612
Hexadecimal	1E16
Armenian	Լ
Hebrew	ל
Babylonian numeral	𒌍
Egyptian hieroglyph	𓎐

30 (thirty) is the natural number following 29 and preceding 31.

In mathematics

30 is a square pyramidal number.

30 is an even, composite, and pronic number. With 2, 3, and 5 as its prime factors, it is a regular number and the first sphenic number, the smallest of the form ${\displaystyle 2\times 3\times r}$, where r is a prime greater than 3. It has an aliquot sum of 42; within an aliquot sequence of thirteen composite numbers (30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0) to the Prime in the 3-aliquot tree. From 1 to the number 30, this is the longest Aliquot Sequence.

It is also:

• A semiperfect number, since adding some subsets of its divisors (e.g., 5, 10 and 15) equals 30. ^[1]
• A primorial. ^[2]
• A Harshad number in decimal. ^[3]
• Divisible by the number of prime numbers (10) below it.
• The largest number such that all coprimes smaller than itself, except for 1, are prime.
• The sum of the first four squares, making it a square pyramidal number. ^[4]
• The number of vertices in the Tutte–Coxeter graph.
• The measure of the central angle and exterior angle of a dodecagon, which is the petrie polygon of the 24-cell.
• The number of sides of a triacontagon, which in turn is the petrie polygon of the 120-cell and 600-cell.
• The number of edges of a dodecahedron and icosahedron, of vertices of an icosidodecahedron, and of faces of a rhombic triacontahedron.
• The sum of the number of elements of a 5-cell: 5 vertices, 10 edges, 10 faces, and 5 cells.
• The Coxeter number of E₈.
• A largely composite number, ^[5] as it has 8 divisors and no smaller number has more than 8 divisors

Furthermore,

In a group G, such that ${\displaystyle |G|=p^{n}\times m}$, where p does not divide m, and has a subgroup of order ${\displaystyle p^{n}}$, 30 is the only number less than 60 that is neither a prime nor of the aforementioned form. Therefore, 30 is the only candidate for the order of a simple group less than 60, in which one needs other methods to specifically reject to eventually deduce said order.^{[ citation needed ]}

The SI prefix for 10³⁰ is Quetta- (Q), and for 10⁻³⁰ (i.e., the reciprocal of 10³⁰) quecto (q). These numbers are the largest and smallest number to receive an SI prefix to date.

In other fields

Thirty is:

• Used (as –30–) to indicate the end of a newspaper (or broadcast) story, a copy editor's typographical notation
• The number of days in the months April, June, September and November (and in unusual circumstances February—see February 30). Although the number of days in a month vary, 30 is used to estimate months elapsing.
• In years of marriage, the pearl wedding anniversary

History and literature

• Age 30 is when Jewish priests traditionally start their service (according to Numbers 4:3).
• One of the rallying cries of the 1960s student/youth protest movement was the slogan, "Don't trust anyone over thirty".
• In The Myth of Sisyphus the French existentialist Albert Camus comments that the age of thirty is a crucial period in the life of a man, for at that age he gains a new awareness of the meaning of time.

References

1. "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
2. "Sloane's A002110 : Primorial numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
3. "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
4. "Sloane's A000330 : Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
5. Sloane, N. J. A. (ed.). "SequenceA067128(Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.