28 (number)

For the card game, see Twenty-eight (card game).

← 27 28 29 →
← 20 21 22 23 24 25 26 27 28 29 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twenty-eight
Ordinal	28th (twenty-eighth)
Factorization	22 × 7
Divisors	1, 2, 4, 7, 14, 28
Greek numeral	ΚΗ´
Roman numeral	XXVIII, xxviii
Binary	111002
Ternary	10013
Senary	446
Octal	348
Duodecimal	2412
Hexadecimal	1C16

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

In mathematics

The number 28 depicted as 28 balls arranged in a triangular pattern with the number of layers of 7

28 as the sum of four nonzero squares.

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]}

In science

• The fourth magic number in physics.

In other fields

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.

References

1. "Sloane's A000396 : Perfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
2. "Sloane's A002088 : Sum of totient function". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
3. "Sloane's A005528 : Størmer numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
4. "Sloane's A001599 : Harmonic or Ore numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
5. "Sloane's A007770 : Happy numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
6. "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
7. "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
8. Sloane, N. J. A. (ed.). "SequenceA045575(Leyland numbers of the second kind)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. "Sloane's A060544 : Centered 9-gonal (also known as nonagonal or enneagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
10. "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
11. "Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
12. "Intersection between the sums of the first positive integers, primes and non primes". mathoverflow.net. Retrieved April 2, 2018.
13. Sloane, N. J. A. (ed.). "SequenceA065577(Number of Goldbach partitions of 10^n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved August 31, 2023.
14. A025368
15. A025359

External links

• Prime Curios! 28 from the Prime Pages