105 (number)

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

In mathematics

105 is:

• the 14th triangular number, a dodecagonal number, and the first Zeisel number. It is the first odd sphenic number and is the product of three consecutive prime numbers. 105 is the double factorial of 7. It is also the sum of the first five square pyramidal numbers.
• 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.
• 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. 105 is the product of the first three odd primes ($3\times 5\times 7$) and is less than the square of the next prime (11) by > 8. Therefore, for $n=105$, n ± 2, ± 4, and ± 8 must be prime (a prime k-tuple). In contrast, n ± 6, ± 10, ± 12, and ± 14 must be composite, making a prime gap on either side.
• 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.
• also a number n for which $n-2^k$ is prime, for $0<k<lo{g}_2(n)$. (This even works up to $k=8$, ignoring the negative sign.)
• the smallest integer such that the factorization of $x^n-1$ over Q includes non-zero coefficients other than $\pm 1$. In other words, the 105th cyclotomic polynomial, Φ₁₀₅, is the first with coefficients other than $\pm 1$.

the number of parallelogram polyominoes with 7 cells.

• the sum of the first 14 natural numbers: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 105.

References

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