Centered nonagonal number

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^[1]

${\displaystyle Nc(n)={\frac {(3n-2)(3n-1)}{2}}.}${\displaystyle Nc(n)={\frac {(3n-2)(3n-1)}{2}}.}

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.^[1]

Thus, the first few centered nonagonal numbers are^[1]

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.^[2] 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.^[3] Pollock's conjecture was confirmed as true in 2023.^[4]

• 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.

• Nonagonal number

• Figurate numbers

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