135 (number)

135 (one hundred [and] thirty-five) is the natural number following 134 and preceding 136.

← 134 135 136 →
← 130 131 132 133 134 135 136 137 138 139 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred thirty-five
Ordinal	135th (one hundred thirty-fifth)
Factorization	33 × 5
Divisors	1, 3, 5, 9, 15, 27, 45, 135
Greek numeral	ΡΛΕ´
Roman numeral	CXXXV
Binary	100001112
Ternary	120003
Senary	3436
Octal	2078
Duodecimal	B312
Hexadecimal	8716

In mathematics

135 is the number of integer partitions of 14, and the number of rooted trees with 15 nodes and height at most 2.^[1] 135 is 5-smooth, since its prime factorization is ${\displaystyle 3^{3}\times 5}$, and a Harshad number in decimal.^[2][3]

Using its own digits, 135 in base-10 can be expressed in operations as the sum of consecutive powers of its digits, and as a sum-product number:

${\displaystyle 135=1^{1}+3^{2}+5^{3}}$^[4]

${\displaystyle 135=(1+3+5)(1\times 3\times 5)}$^[5]

135 is the number of degrees in the internal angle of a regular eight-sided octagon, and the number of nodes inside a regular nonagon from the intersection of its diagonals and sides.^[6] Also:

• the Euler totient of 135 is 72: the degrees of a central angle in a regular pentagon.^[7]
• the arithmetic mean of the divisors of 135 is 30: the degrees in a central angle of a regular dodecagon.

While the central angle of a regular octagon is 135 ÷ 3 = 45 degrees, 4.5 is the harmonic mean of all eight divisors of 135.

The aliquot sum of 135 is 105, which is the 14th triangular number, or equivalently the sum of the first fourteen non-zero positive integers.^[8][9]

There are 135 total Krotenheerdt k-uniform tilings for k < 8, with no other such tilings for higher k.^[10]

There are a total of 135 primes between 1,000 and 2,000.

${\displaystyle 135=11n^{2}+11n+3}$ for ${\displaystyle n=3}$ is a polynomial that plays an essential role in Apéry's proof that ${\displaystyle \zeta (3)}$ is irrational.^{[citation needed]}

In other fields

• The year AD 135 or 135 BC.
• 135 AH is a year in the Islamic calendar that corresponds to 752–753 CE.
• 135 Hertha is a large main belt asteroid which orbits among the Nysa asteroid family.
• 135 film, the cartridge version of 35mm photographic film, used widely in still photography.
• The Canon FD 135 mm lens.
• In astrology, when two planets are 135 degrees apart, they are in an astrological aspect called a sesquiquadrate. The aspect was first used by Johannes Kepler.
• Sonnet 135 by William Shakespeare.
• Municipal District of Peace No. 135, a municipal district in northwest Alberta, Canada.
• Enoch Cree Nation 135 Indian reserve in Alberta, Canada, is home to the Enoch Cree Nation.
• The EZ 135 Drive removable hard disk drive introduced by SyQuest Technology in 1995.
• OC-135B Open Skies United States Air Force observation aircraft flies unarmed observation flights over nations of the Treaty on Open Skies.

See also

• List of highways numbered 135
• United Nations Security Council Resolution 135
• United States Supreme Court cases, Volume 135

References

1. Sloane, N. J. A. (ed.). "Sequence A000041 (The number of partitions of n (the partition numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05.
2. Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05.
3. Sloane, N. J. A. (ed.). "Sequence A051037 (5-smooth numbers, i.e., numbers whose prime divisors are all less than or equal to five.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05.
4. "Sloane's A032799: Numbers n such that n equals the sum of its digits raised to the consecutive powers (1,2,3,...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2019-12-08.
5. "Sloane's A038369 : Numbers n such that n = (product of digits of n) * (sum of digits of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
6. Sloane, N. J. A. (ed.). "Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
7. Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-06.
8. Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-06.
9. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-06.
10. Sloane, N. J. A. (ed.). "Sequence A068600 (Number of n-uniform tilings having n different arrangements of polygons about their vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-09.