105 (number)

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, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the first three odd primes (${\displaystyle 3\times 5\times 7}$) and less than the square of the next prime (11) by > 8, for ${\displaystyle n=105}$, 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 ${\displaystyle n-2^{k}}$ is prime, for ${\displaystyle 0<k<log_{2}(n)}$. (This even works up to ${\displaystyle k=8}$, ignoring the negative sign.)

105 is the smallest integer such that the factorization of ${\displaystyle x^{n}-1}$ over Q includes non-zero coefficients other than ${\displaystyle \pm 1}$. In other words, the 105th cyclotomic polynomial, Φ₁₀₅, is the first with coefficients other than ${\displaystyle \pm 1}$.

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

• The atomic number of dubnium.

105 is also:

• A Shimano Road groupset since 1984

• List of highways numbered 105

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134