Centered nonagonal number
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A centered nonagonal 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 is given by the formula
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.
Thus, the first few centered nonagonal numbers are
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, 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 2p-1 is a Mersenne prime.
In 1850, Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven.
- regular nonagonal number