15 (number)

15 is:

• The eighth composite number and the sixth semiprime and the first odd and fourth discrete semiprime;[1] its proper divisors are 1, 3, and 5, so the first of the form (3.q),[2] where q is a higher prime.
• a deficient number, a lucky number, a bell number (i.e., the number of partitions for a set of size 4), [3] a pentatope number, [4] and a repdigit in binary (1111) and quaternary (33). In hexadecimal, and higher bases, it is represented as F.
• with an aliquot sum of 9; within an aliquot sequence of three composite numbers (15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
• the second member of the first cluster of two discrete semiprimes (14, 15); the next such cluster is (21, 22).
• the first number to be polygonal in 3 ways: it is a triangular number, a hexagonal number,[5] and pentadecagonal number.[6]
• a centered tetrahedral number.
• the number of partitions of 7.
• the smallest number that can be factorized using Shor's quantum algorithm.
• the magic constant of the unique order-3 normal magic square.
• the number of supersingular primes.
• the smallest positive number that can be expressed as the difference of two positive squares in more than one way:[7] ${\displaystyle 4^{2}-1^{2}}$ or ${\displaystyle 8^{2}-7^{2}}$ (see image).

Furthermore,

• 15's prime factors, (3 and 5), form the first twin-prime pair.
• The first 15 superabundant numbers are the same as the first 15 colossally abundant numbers.
• In decimal, 15 contains the digits 1 and 5 and is the result of adding together the integers from 1 to 5 (1 + 2 + 3 + 4 + 5 = 15). The only other number with this property (in decimal) is 27.
• There are 15 truncatable primes that are both right-truncatable and left-truncatable:

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (sequence A020994 in the OEIS)

• There are 15 perfect matchings of the complete graph K₆ and 15 rooted binary trees with four labeled leaves, both of these being among the types of objects counted by double factorials.
• With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet (11, 13, 17, 19).
• If a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems.
• 15 is the product of distinct Fermat primes, 3 and 5; hence, a regular pentadecagon is constructible with a compass and unmarked straightedge, and ${\displaystyle \cos {\frac {\pi }{15}}}$ is expressible in terms of square roots.
• There are 15 monohedral convex pentagonal tilings, with eight being edge-to-edge.
• There are 15 regular and semiregular tilings when infinite (improper) apeirogonal forms are counted: three are regular (with one self-dual), eight are semiregular (with one chiral), and four are apeirogonal (from a total of 8, in-which 4 are duplicates).
• Full icosahedral symmetry contains 15 mirror planes (2-fold axes). Specifically, the symmetry order for both the regular icosahedron and regular dodecahedron (which is made of regular pentagons) is 120: equal to sum of the first 15 integers, and the factorial of 5, wherein the sum of the first 5 integers itself is 15. Expressed mathematically:

  ${\displaystyle \sum _{i=1}^{15}i=120}$, while ${\displaystyle \sum _{i=1}^{5}i=15}$, and ${\displaystyle 5!=120}$.

• There are 15 Archimedean solids and 15 Catalan solids when enantiomorphic forms are counted separately.
• There are 15 regular honeycombs in hyperbolic 3-space: four are compact, and 11 are paracompact.[8]
• It is the smallest non-trivial Surprise Number. When 15 is partitioned into 2 parts of digits- smaller part 1 and larger part 5, adding numbers from smaller 1 to larger 5 = 1 + 2 + 3 + 4 + 5 = 15 = Original Number.

• The atomic number of phosphorus.
• 15 Eunomia is the largest Eunomian asteroid in the inner asteroid belt.

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 91–93

• Clewett, James. "15: f in hexadecimal". Numberphile. Brady Haran. Archived from the original on 2013-05-16. Retrieved 2013-04-02. – discussing hexadecimals
• Bowley, Roger. "15: Bumfit". Numberphile. Brady Haran. Archived from the original on 2013-05-16. Retrieved 2013-04-01. – discussing the Celtic number as used in Lincolnshire