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135 (number)

From Wikipedia, the free encyclopedia
← 134 135 136 →
Cardinalone hundred thirty-five
Ordinal135th
(one hundred thirty-fifth)
Factorization33 × 5
Divisors1, 3, 5, 9, 15, 27, 45, 135
Greek numeralΡΛΕ´
Roman numeralCXXXV
Binary100001112
Ternary120003
Senary3436
Octal2078
DuodecimalB312
Hexadecimal8716

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

In mathematics

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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 , 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:

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[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:

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.

for is a polynomial that plays an essential role in Apéry's proof that is irrational.[citation needed]

In other fields

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See also

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References

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  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.