A centered nonagonal number is a centered

Centered number

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides...

 figurate number

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...

 that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for n is given by the formula


Multiplying the (n - 1)th 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...

 by 9 and then adding 1 yields the nth centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number is also a centered nonagonal number.

Thus, the first few centered nonagonal numbers are

1, 10

10 (number)

10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...

, 28

28 (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....

, 55

55 (number)

55 is the natural number following 54 and preceding 56.-Albania:*Gazeta 55, a newspaper*Constitution law 55, a law during Communist Albania.-Mathematics:...

, 91

91 (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...

, 136, 190

190 (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...

, 253, 325, 406, 496

496 (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:...

, 595, 703, 820, 946.

Note the following perfect numbers that are in the list:

The 3rd centered nonagonal number is 7 x 8 / 2 = 28, and the 11th is 31 x 32 / 2 = 496.

Proceeding further: the 43rd is 127 x 128 / 2 = 8128

8128 (number)

8128 is the natural number following 8127 and preceding 8129.It is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 127, 27 − 1, with 26 · yielding 8128...

, and the 2731st is 8191 x 8192 / 2 = 33,550,336.

Except for 6, all even perfect numbers are also centered nonagonal numbers, with formula

where 2^p-1 is a Mersenne prime

Mersenne prime

In 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.\,...

.

In 1850, Pollock

Sir Frederick Pollock, 1st Baronet

Sir Frederick Pollock, 1st Baronet PC , was a British lawyer and Tory politician.-Background and education:...

claimed (without proof) that every natural number is the sum of at most eleven centered nonagonal numbers, a conjecture which has been neither proven nor disproven.