360 (number)

This article is about the number. For the year in the 4th century, see AD 360. For other uses, see 360 (disambiguation).

← 359 360 361 →
List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	three hundred sixty
Ordinal	360th (three hundred sixtieth)
Factorization	23 × 32 × 5
Divisors	1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
Greek numeral	ΤΞ´
Roman numeral	CCCLX
Binary	1011010002
Ternary	1111003
Senary	14006
Octal	5508
Duodecimal	26012
Hexadecimal	16816

The surface of the compound of five cubes consists of 360 triangles.

360 (three hundred [and] sixty) is the natural number following 359 and preceding 361.

In mathematics

• 360 is a highly composite number ^[1] and one of only seven numbers such that no number less than twice as much has more divisors; the others are 1, 2, 6, 12, 60, and 2520 (sequence A072938 in the OEIS ).

• 360 is also a superior highly composite number, a colossally abundant number, a refactorable number, a 5-smooth number, and a Harshad number in decimal since the sum of its digits (9) is a divisor of 360.

• 360 is divisible by the number of its divisors (24), and it is the smallest number divisible by every natural number from 1 to 10, except 7. Furthermore, one of the divisors of 360 is 72, which is the number of primes below it.

• 360 is the sum of twin primes (179 + 181) and the sum of four consecutive powers of three (9 + 27 + 81 + 243).

• The sum of Euler's totient function φ(x) over the first thirty-four integers is 360.

• 360 is a triangular matchstick number. ^[2]

• 360 is the product of the first two unitary perfect numbers: ^[3] ${\displaystyle 60\times 6=360.}$

• There are 360 even permutations of 6 elements. They form the alternating group A₆.

A circle is divided into 360 degrees for angular measurement. 360° = 2π  rad is also called a round angle. This unit choice divides round angles into equal sectors measured in integer rather than fractional degrees. Many angles commonly appearing in planimetrics have an integer number of degrees. For a simple non-intersecting polygon, the sum of the internal angles of a quadrilateral always equals 360 degrees.

Integers from 361 to 369

361

${\displaystyle 361=19^{2},}$ centered triangular number, ^[4] centered octagonal number, centered decagonal number, ^[5] member of the Mian–Chowla sequence, ^[6] and the number of positions on a standard 19 × 19 Go board.

362

${\displaystyle 362=2\times 181=\sigma _{2}(19)}$: sum of squares of divisors of 19, ^[7] Mertens function returns 0, ^[8] nontotient, noncototient. ^[9]

363

Main article: 363 (number)

364

${\displaystyle 364=2^{2}\times 7\times 13}$, tetrahedral number, ^[10] sum of twelve consecutive primes (11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53), Mertens function returns 0, ^[11] nontotient.

It is a repdigit in bases three (111111), nine (444), twenty-five (EE), twenty-seven (DD), fifty-one (77), and ninety (44); the sum of six consecutive powers of three (1 + 3 + 9 + 27 + 81 + 243); and the twelfth non-zero tetrahedral number. ^[12]

365

Main article: 365 (number)

366

${\displaystyle 366=2\times 3\times 61,}$ sphenic number, ^[13] Mertens function returns 0, ^[14] noncototient, ^[15] number of complete partitions of 20, ^[16] 26-gonal and 123-gonal. There are also 366 days in a leap year.

367

367 is a prime number, Perrin number, ^[17] happy number, prime index prime and a strictly non-palindromic number.

368

${\displaystyle 368=2^{4}\times 23.}$ It is also a Leyland number. ^[18]

369

Main article: 369 (number)

References

1. Sloane, N. J. A. (ed.). "SequenceA002182(Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
2. Sloane, N. J. A. (ed.). "SequenceA045943(Triangular matchstick numbers: a(n) is 3*n*(n+1)/2)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. Sloane, N. J. A. (ed.). "SequenceA002827(Unitary perfect numbers: numbers k such that usigma(k) - k equals k.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-11-02.
4. "Centered Triangular Number". mathworld.wolfram.com.
5. Sloane, N. J. A. (ed.). "SequenceA062786(Centered 10-gonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-22.
6. Sloane, N. J. A. (ed.). "SequenceA005282(Mian-Chowla sequence)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-22.
7. Sloane, N. J. A. (ed.). "SequenceA001157(a(n) = sigma_2(n): sum of squares of divisors of n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
8. Sloane, N. J. A. (ed.). "SequenceA028442(Numbers k such that Mertens's function M(k) (A002321) is zero)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. "Noncototient". mathworld.wolfram.com.
10. Sloane, N. J. A. (ed.). "SequenceA000292(Tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-22.
11. Sloane, N. J. A. (ed.). "SequenceA028442(Numbers k such that Mertens's function M(k) (A002321) is zero)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
12. Sloane, N. J. A. (ed.). "SequenceA000292(Tetrahedral (or triangular pyramidal) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
13. "Sphenic number". mathworld.wolfram.com.
14. Sloane, N. J. A. (ed.). "SequenceA028442(Numbers k such that Mertens's function M(k) (A002321) is zero)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
15. "Noncototient". mathworld.wolfram.com.
16. Sloane, N. J. A. (ed.). "SequenceA126796(Number of complete partitions of n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
17. "Parrin number". mathworld.wolfram.com.
18. Sloane, N. J. A. (ed.). "SequenceA076980". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.

Sources

• Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (p. 152). London: Penguin Group.

External links

• Media related to 360 (number) at Wikimedia Commons