28 (number)

28 (twenty-eight) is the natural number following 27 and preceding 29.

Twenty-eight is a composite number and the second perfect number as it is the sum of its proper divisors: ${\displaystyle 1+2+4+7+14=28}$. As a perfect number, it is related to the Mersenne prime 7, since ${\displaystyle 2^{3-1}\times (2^{3}-1)=28}$. The next perfect number is 496, the previous being 6.^[1]

Though perfect, 28 is not the aliquot sum of any other number other than itself; thus, it is not part of a multi-number aliquot sequence. The next perfect number is 496.

Twenty-eight is the sum of the totient function for the first nine integers.^[2]

Since the greatest prime factor of ${\displaystyle 28^{2}+1=785}$ is 157, which is more than 28 twice, 28 is a Størmer number.^[3]

Twenty-eight is a harmonic divisor number,^[4] a happy number,^[5] the 7th triangular number,^[6] a hexagonal number,^[7] a Leyland number of the second kind^[8] (${\displaystyle 2^{6}-6^{2}}$), and a centered nonagonal number.^[9]

It appears in the Padovan sequence, preceded by the terms 12, 16, 21 (it is the sum of the first two of these).^[10]

It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its decimal digits: 2, 8, 10, 18, 28...^[11]

There are 28 convex uniform honeycombs.

Twenty-eight is the only positive integer that has a unique Kayles nim-value.

Twenty-eight is the only known number that can be expressed as a sum of the first positive integers (${\displaystyle 1+2+3+4+5+6+7}$), a sum of the first primes (${\displaystyle 2+3+5+7+11}$), and a sum of the first nonprimes (${\displaystyle 1+4+6+8+9}$), and it is unlikely that any other number has this property.^[12]

There are twenty-eight oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere.^{[citation needed]}

There are 28 non-equivalent ways of expressing 1000 as the sum of two prime numbers.^[13]

Twenty-eight is the smallest number that can be expressed as the sum of four nonzero squares in (at least) three ways: ${\displaystyle 5^{2}+1^{2}+1^{2}+1^{2}}$, ${\displaystyle 4^{2}+2^{2}+2^{2}+2^{2}}$ or ${\displaystyle 3^{2}+3^{2}+3^{2}+1^{2}}$ (see image).^[14][15]

• The fourth magic number in physics.

Twenty-eight is:

• Deriving from the 29.46 year period of Saturn's revolution around the Sun, the 28-year cycle as well as its subdivisions by 14 and 7 are supposed in astrology to mark significant turning points or sections in the course of a person's development in life. Thus, the number 28 has special significance in the culture of religious sects such as the Kadiri and the Mevlevi dervishes. The 28-beat metric pattern often used in the music compositions accompanying the main part of the Mevlevi sema ritual is called the "Devri kebir", meaning the "Big Circle" and is a reference to above astronomical facts about the year and the Saturn year.
• In Quebec, François Pérusse, in one of his best-selling Album du peuple made a parody of Wheel of Fortune in which all of the letters picked by the contestant were present 28 times. As a result, 28 became an almost mythical number used by many Quebec youths, the phrase "Y'en a 28" (There are 28 [Letters]) became a running gag still used and recognized more than 15 years later.
• Approximately the number of grams in an ounce, and used as such in the illegal drug trade.

• Prime Curios! 28 from the Prime Pages