105 (number)
| ||||
|---|---|---|---|---|
| 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 | |||
| Binary | 11010012 | |||
| Ternary | 102203 | |||
| Quaternary | 12214 | |||
| Quinary | 4105 | |||
| Senary | 2536 | |||
| Octal | 1518 | |||
| Duodecimal | 8912 | |||
| Hexadecimal | 6916 | |||
| Vigesimal | 5520 | |||
| Base 36 | 2X36 | |||
105 (one hundred [and] five) is the natural number following 104 and preceding 106.
In mathematics
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 ([math]\displaystyle{ 3\times5\times7 }[/math]) and less than the square of the next prime (11) by > 8, for [math]\displaystyle{ n=105 }[/math], n ± 2, ± 4, and ± 8 must be prime, and n ± 6, ± 10, ± 12, and ± 14 must be composite (prime gap).[clarification needed]
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 [math]\displaystyle{ n - 2^k }[/math] is prime, for [math]\displaystyle{ 0 \lt k \lt log_2(n) }[/math]. (This even works up to [math]\displaystyle{ k = 8 }[/math], ignoring the negative sign.)
105 is the smallest integer such that the factorization of [math]\displaystyle{ x^n-1 }[/math] over Q includes non-zero coefficients other than [math]\displaystyle{ \pm 1 }[/math]. In other words, the 105th cyclotomic polynomial, Φ105, is the first with coefficients other than [math]\displaystyle{ \pm 1 }[/math].
105 is the number of parallelogram polyominoes with 7 cells.[4]
In science
- The atomic number of dubnium.
In other fields
105 is also:
- A Shimano Road groupset since 1984
See also
- List of highways numbered 105
References
- Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134
- ↑ "Sloane's A051624 : 12-gonal numbers". OEIS Foundation. https://oeis.org/A051624.
- ↑ "Sloane's A051015 : Zeisel numbers". OEIS Foundation. https://oeis.org/A051015.
- ↑ "Sloane's A006882 : Double factorials". OEIS Foundation. https://oeis.org/A006882.
- ↑ Sloane, N. J. A., ed. "Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))". OEIS Foundation. https://oeis.org/A006958.
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