105 (one hundred [and] five) is the natural number following 104 and preceding 106.
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. 105 is the product of the first three odd primes ( 3 × × --> 5 × × --> 7 {\displaystyle 3\times 5\times 7} ) and is less than the square of the next prime (11) by > 8. Therefore, for n = 105 {\displaystyle 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.
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 n − − --> 2 k {\displaystyle n-2^{k}} is prime, for 0 < k < l o g 2 ( n ) {\displaystyle 0<k<log_{2}(n)} . (This even works up to k = 8 {\displaystyle k=8} , ignoring the negative sign.)
105 is the smallest integer such that the factorization of x n − − --> 1 {\displaystyle x^{n}-1} over Q includes non-zero coefficients other than ± ± --> 1 {\displaystyle \pm 1} . In other words, the 105th cyclotomic polynomial, Φ105, is the first with coefficients other than ± ± --> 1 {\displaystyle \pm 1} .
105 is the number of parallelogram polyominoes with 7 cells.[4]
105 is also:
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