TWO NEW KINDS OF NUMBERS AND RELATED DIVISIBILITY RESULTS

We mainly introduce two new kinds of numbers given by R-n = Sigma(n)(k=0) ((n)(k)) ((n + k)(k)) 1/2k - 1 (n = 0,1,2,...), S-n = Sigma(n)(k=0) ((n)(k))(2) ((2k)(k)) (2k + 1) (n = 0,1,2,...). We find that such numbers have many interesting arithmetic properties. For example, if p 1 (mod 4) is a prime with p = x(2 )+ y(2) (where x 1 (mod 4) and y 0 (mod 2)), then R(p-1)/2 p - (-1)((p-1)/4)2x ( mod p(2)). Also, 1/n(2) Sigma(n-1)(k=0) S-k is an element of Z and 1/n Sigma(n-1)(k=0) S-k(x) is an element of Z[x] for all n = 1,2,..., where S-k(x) = Sigma(k)(j=0) ((k)(j))(2) ((2j)(j)) (2j + 1)x(j). For any positive integers a and n, we show that, somewhat surprisingly, 1/n(2) Sigma(n-1)(k=0)(2k + 1) ((n - 1)(k))(a) ((-n - 1)(k))(a) is an element of Z and 1/n Sigma(n-1)(k=0) ((n - 1)(k))(a) ((-n - 1)(k))(a)/4k(2) - 1 is an element of Z. We also solve a conjecture of V. J. W. Guo and J. Zeng, and pose several conjectures for further research.