Centered nonagonal number

File:Centered nonagonal number.svg

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<ref name="oeis">Template:Cite OEIS</ref>

<math>Nc(n) = \frac{(3n-2)(3n-1)}{2}.</math>

Multiplying the (n - 1)th triangular number 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 (the 1st, 4th, 7th, etc.) is also a centered nonagonal number.<ref name="oeis"/>

Thus, the first few centered nonagonal numbers are<ref name="oeis"/>

1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946.

The list above includes the perfect numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime.<ref>Template:Citation.</ref> Since every Mersenne prime greater than 3 is congruent to 1 modulo 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.<ref>Template:Citation.</ref> Pollock's conjecture was confirmed as true in 2023.<ref>Template:Cite journal</ref>

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.

See also

• Nonagonal number