105 (number)

105 1, 3, 5, 7, 15, 21, 35, 105 105 (one hundred [and] five) is the natural number following 104 and preceding 106. In mathematics 105 is the 14th triangular number,A000217 - OEIS https://oeis.org/A000217 2024-11-28 oeis.org a dodecagonal number,https://oeis.org/A051624 Sloane's A051624 : 12-gonal numbers The On-Line Encyclopedia of Integer Sequences OEIS Foundation 2016-05-27 and the first Zeisel number.https://oeis.org/A051015 Sloane's A051015 : Zeisel numbers The On-Line Encyclopedia of Integer Sequences OEIS Foundation 2016-05-27 It is the first odd sphenic number. 105 is the double factorial of 7.https://oeis.org/A006882 Sloane's A006882 : Double factorials The On-Line Encyclopedia of Integer Sequences OEIS Foundation 2016-05-27 It is also the sum of the first five square pyramidal numbers. It is 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. It 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 () and is less than the square of the next prime (11) by more than 8. Therefore, for , 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 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 a number n for which is prime, for . (This 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 . There are 105 parallelogram polyominoes with 7 cells.A006958 Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused) References Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134 1 Category:Integers