number around a central point; therefore, the nth centered k-gonal number can be mathematically represented by The difference of the n-th and the (n+1)-th consecutive centered k-gonal numbers is k(2n+1). The n-th centered k-gonal number is equal to the n-th regular k-gonal number plus (n-1)2. Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula: which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc. Whereas a prime number p cannot be a polygonal number (except the trivial case, i.e. each p is the second p-gonal number), many centered polygonal numbers are primes. In fact, if k ≥ 3, k ≠ 8, k ≠ 9, then there are infinitely many centered k-gonal numbers which are primes (assuming the Bunyakovsky conjecture). (Since all centered octagonal numbers are also square numbers, and all centered nonagonal numbers are also triangular numbers (and not equal to 3), thus both of them cannot be prime numbers.) Sum of reciprocals The sum of reciprocals for the centered k-gonal numbers is , if k ≠ 8 , if k = 8 References : Fig. M3826 Figurate numbers