A centered nonagonal number (or centered enneagonal number) is a centered figurate number 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'' layers is given by the formula :

Nc(n) = \frac.

Multiplying the (''n'' - 1)th

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...

by 9 and then adding 1 yields the ''n''th centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number. Thus, the first few centered nonagonal numbers are : 1, 10, 28, 55, 91, 136,

190 Year 190 (CXC) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Sura (or, less frequently, year 943 ''Ab urbe condita'') ...

, 253, 325, 406,

496 __NOTOC__ Year 496 ( CDXCVI) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Paulus without colleague (or, less frequently, ye ...

, 595, 703, 820, 946. The list above includes the

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...

s 28 and 496. All even perfect numbers are triangular numbers whose index is an odd

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...

. Since every Mersenne prime greater than 3 is congruent to 1

modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...

3, it follows that every even perfect number greater than 6 is a centered nonagonal number. In 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven..

Congruence Relations

*All centered nonagonal numbers are congruent to 1 mod 3. **Therefore the sum of any 3 centered nonagonal numbers and the difference of any two centered nonagonal numbers are divisible by 3.

References

{{DEFAULTSORT:Centered Nonagonal Number Figurate numbers

Congruence Relations

See also

References