Since deg(g) = deg(f) [greater than or equal to] 2, we see that 0 < [([D.sub.n]).sup.k] < 1 for a sufficiently

positive integer n.

Algorithm 2 Let n be a nonnegative integer, [lambda] [member of] C, and k be a

positive integer. This algorithm will return the numbers Wn ([lambda]) recursively with the help of the procedure W_APOSTOL_TYPE_NUM given by Algorithm 1.

Let p be a prime and e [greater than or equal to] 2, k [greater than or equal to] 2 be two

positive integers. Let h be the unique

positive integer such that [k.sup.h-1] < e [less than or equal to] [k.sup.h].

Let N be a

positive integer. Then N is a trapezoidal number if and only if N is not of the form [2.sup.i] for all i [member of] N [union] {0}.

From (37) and (46) we see that [[??].sub.k](n) = [S.sub.k](n) for each n = 1, 2, ..., and thus the polynomial identity (45) with respect to the

positive integer variable n can be extended to the real variable x instead of n; that is, (45) is identically satisfied (on R) if we replace [S.sub.l](n) by [[??].sub.i](x) (with l = 2k + 1 and l = 2i).

where p, q are any two

positive integers with p [greater than or equal to] q.

If, for some

positive integer k and [alpha] [greater than or equal to] 0, ([[tau].sub.n]) is [(A).sup.(k)](C, [alpha]) summable to s and

Since M is [pi]-Rickart, [r.sub.M] ([f.sup.n]) = eM for some

positive integer n and [e.sup.2] = e [member of] S.

where q is any

positive integer such that m [greater than or equal to] q.

We recall that the LCM of two whole numbers a and b is the smallest

positive integer having both a and b as factors (see Burton, 2002, p.

The Beal Conjecture states that the only solutions to the equation Ax + By = Cz, when A, B, C, are

positive integers, and x, y, and z are

positive integers greater than two, are those in which A, B, and C have a common factor.

Inspired by the above conclusions, in this paper, we study the hybrid mean value properties of the Smarandache kn-digital sequence with SL(n) function and divisor function d(n), where SL(n) is defined as the smallest

positive integer k such that n|[1, 2,...,k], that is SL(n) = min{k : k [member of] N, n|[1, 2,..., k]}.