Hexagonal number
Encyclopedia
A hexagonal number is a figurate number
. The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side.
The formula for the nth hexagonal number
The first few hexagonal numbers are:
Every hexagonal number is a triangular number
, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root
in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9".
Every even perfect number
is hexagonal, given by the formula
The largest number that cannot be written as a sum of at most four hexagonal numbers is 130
. Adrien-Marie Legendre
proved in 1830 that any integer
greater than 1791 can be expressed in this way.
Hexagonal numbers can be rearranged into rectangular numbers of size n by (2n−1).
Hexagonal numbers should not be confused with centered hexagonal number
s, which model the standard packaging of Wiener sausage
s. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
If n is an integer, then x is the nth hexagonal number. If n is not an integer, then x is not hexagonal.
where the empty sum is taken to be 0.
Figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...
. The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side.
The formula for the nth hexagonal number
The first few hexagonal numbers are:
- 1, 6, 1515 (number)15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name....
, 2828 (number)28 is the natural number following 27 and preceding 29.-In mathematics:It is a composite number, its proper divisors being 1, 2, 4, 7, and 14....
, 4545 (number)45 is the natural number following 44 and followed by 46.- In mathematics :Forty-five is a triangular number, a hexagonal and 16-gonal number, a Kaprekar number, and a Harshad number....
, 6666 (number)66 is the natural number following 65 and preceding 67.Usages of this number include:-Mathematics:*66 is a sphenic number, a triangular number, a hexagonal number, and a semi-meandric number...
, 9191 (number)91 is the natural number following 90 and preceding 92.-In mathematics:Ninety-one is the twenty-seventh distinct semiprime and the second of the form...
, 120120 (number)120 is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120. The number 100, now known commonly as "one hundred" was then known as "the small hundred". It is also known as...
, 153153 (number)One hundred fifty-three is the natural number following one hundred fifty-two and preceding one hundred fifty-four.-Mathematical properties:...
, 190190 (number)190 is the natural number following 189 and preceding 191.-In mathematics:* 190 is an even number* 190 is a centered nonagonal number* 190 is a composite number* 190 is a deficient number, as 170 is less than 190...
, 231, 276, 325, 378, 435, 496496 (number)Four hundred [and] ninety-six is the natural number following four hundred [and] ninety-five and preceding four hundred [and] ninety-seven.-In mathematics:...
, 561, 630, 703, 780, 861, 946.
Every hexagonal number is a triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root
Digital root
The digital root of a number is the value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum...
in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9".
Every even perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
is hexagonal, given by the formula
- where Mp is a Mersenne primeMersenne primeIn mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
. No odd perfect numbers are known, hence all known perfect numbers are hexagonal. - For example, the 2nd hexagonal number is 2x3 = 6; the 4th is 4x7 = 28; the 16th is 16x31 = 496; and the 64th is 64x127 = 8128.
The largest number that cannot be written as a sum of at most four hexagonal numbers is 130
130 (number)
130 is the natural number following 129 and preceding 131.-In mathematics:130 is a sphenic number. It is a noncototient since there is no answer to the equation x - φ = 130....
. Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
proved in 1830 that any integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
greater than 1791 can be expressed in this way.
Hexagonal numbers can be rearranged into rectangular numbers of size n by (2n−1).
Hexagonal numbers should not be confused with centered hexagonal number
Centered hexagonal number
A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice....
s, which model the standard packaging of Wiener sausage
Wiener sausage
In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion...
s. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
Test for hexagonal numbers
One can efficiently test whether a positive integer x is an hexagonal number by computingIf n is an integer, then x is the nth hexagonal number. If n is not an integer, then x is not hexagonal.
Other Properties
The nth number of the hexagonal sequence can also be expressed by using Sigma notation aswhere the empty sum is taken to be 0.
External links
- MathworldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
entry on Hexagonal Number