105 (number)

← 104 105 106 →
← 100 101 102 103 104 105 106 107 108 109 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred five
Ordinal	105th (one hundred fifth)
Factorization	3 × 5 × 7
Divisors	1, 3, 5, 7, 15, 21, 35, 105
Greek numeral	ΡΕ´
Roman numeral	CV, cv
Binary	11010012
Ternary	102203
Senary	2536
Octal	1518
Duodecimal	8912
Hexadecimal	6916

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

In mathematics

105 is:

• the 14th triangular number, ^[1] a dodecagonal number, ^[2] and the first Zeisel number. ^[3] It is the first odd sphenic number and is the product of three consecutive prime numbers. 105 is the double factorial of 7. ^[4] 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 (${\displaystyle 3\times 5\times 7}$) and is less than the square of the next prime (11) by > 8. Therefore, for ${\displaystyle n=105}$, 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.
• 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 ${\displaystyle n-2^{k}}$ is prime, for ${\displaystyle 0<k<log_{2}(n)}$. (This even works up to ${\displaystyle k=8}$, ignoring the negative sign.)
• 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 ${\displaystyle \pm 1}$.

the number of parallelogram polyominoes with 7 cells. ^[5]

• 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

1. "A000217 - OEIS". oeis.org. Retrieved 2024-11-28.
2. "Sloane's A051624 : 12-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
3. "Sloane's A051015 : Zeisel numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
4. "Sloane's A006882 : Double factorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
5. Sloane, N. J. A. (ed.). "SequenceA006958(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.