A property of the Rayleigh function

It would be of some interest to know exactly which primes divide the numerator or denominator of o-2n(v), if v is rational and oa2n(v) is reduced to its lowest terms. Restricting ourselves to the prime 2, and using induction on (1.1), we shall find in this paper the exact power of 2 dividing o-2n(v) when v is a rational number a/b, a odd and b even. This will extend the results of [2 ] where the same problem was solved for v one half an odd integer. In this paper we shall also prove some congruences (mod 2), (mod 4), and (mod 8). We note that this type of problem has been considered for other sequences of rational numbers. For the well-known Bernoulli numbers B2n, for example, it has been proved [1] that