A divisibility property of binomial coefficients viewed as an elementary sieve

The triangular array of binomial coefficients 0 1 2 3 0 1 1 1 1 2 1 2 1 3 1 3 3 1 … is said to have undergone a j -shift if the r -th row of the triangle is shifted r j units to the right ( r = 0 , 1 , 2 , … ) . Mann and Shanks have proved that in a 2-shifted array a column number c > 1 is prime if and only if every entry in the c -th column is divisible by its row number. Extensions of this result to j -shifted arrays where j > 2 are considered in this paper. Moreover, an analog of the criterion of Mann and Shanks [2] is given which is valid for arbitrary arithmetic progressions.