105 (number)

105 (one hundred [and] five) is the natural number following 104 and preceding 106.

← 104 105 106 →
Cardinalone hundred five
Ordinal105th
(one hundred fifth)
Factorization3 × 5 × 7
Divisors1, 3, 5, 7, 15, 21, 35, 105
Greek numeralΡΕ´
Roman numeralCV
Binary11010012
Ternary102203
Senary2536
Octal1518
Duodecimal8912
Hexadecimal6916

In mathematicsEdit

105 is a triangular number, a dodecagonal number[1] and the first Zeisel number.[2] It is the first odd sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7.[3] It is also the sum of the first five square pyramidal numbers.

105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such numbers less than a thousand are 9, 15, 195 and 825.

105 is also the middle of the only prime sextuplet (97, 101, 103, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the 1st 3 odd primes (3,5,7), and less than the square of the next prime, 11, by > 8, for n=105, n ± 2, ± 4, and ± 8 must be prime and n ± 6, ± 10, ± 12, and ± 14 must be composite (prime gap).

105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83 and 97. The distinct prime factors of 105 add up to 15, and so do those of 104, hence the two numbers form a Ruth-Aaron pair under the first definition.

105 is also a number n for which   is prime, for  . (This even works up to  , ignoring the negative sign.)

105 is the smallest integer such that the factorization of   over Q includes non-zero coefficients other than  . In other words, the 105th cyclotomic polynomial, Φ105, is the first with coefficients other than  .

105 is the number of parallelogram polyominoes with 7 cells.[4]

In scienceEdit

In other fieldsEdit

105 is also:

See alsoEdit

ReferencesEdit

  1. ^ "Sloane's A051624 : 12-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
  2. ^ "Sloane's A051015 : Zeisel numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
  3. ^ "Sloane's A006882 : Double factorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.