## On the Pseudo-Smarandache Function

### By

### J. Sándor

### Babes-Bolyai Univ., 3400 Cluj, Romania

Kashihara[2] defined the "Pseudo-Smarandache" function Z by

Properties of this function have been studied in [1], [2] etc.

1. By answering a question by C. Ashbacher, Maohua Le proved that S(Z(n)) - Z(S(n)) changes signs
infinitely often. Put

We will prove first that

and

implying (1) . We note that if the equation S(Z(n)) = Z(S(n)) has infinitely many solutions, then
clearly the lim inf in (1) is 0, otherwise is 1, since

| S(Z(n)) - Z(S(n)) | >= 1,
S(Z(n)) - Z(S(n)) being an integer.

This inequality is best possible for even n, since Z(2^{k}) = 2^{k+1} - 1. We note that
for odd n, we have Z(n) <= n - 1, and this is best possible for odd n, since Z(p) = p-1 for prime p.

By

On the other hand, by Z(Z(n)) <= 2Z(n) - 1 and (3), we have

Indeed, in [1] it was proved that Z(2p) = p-1 for a prime p congruent to 1 modulo 4. Since Z(p) = p-1,
this proves relation (7).

On the other hand, let n = 2^{k}. Since Z(2^{k}) = 2^{k+1} - 1 and
Z(2^{k+1}) = 2^{k+2} - 1, clearly
Z(2^{k+1}) - Z(2^{k}) = 2^{k+1} -> infinity as k -> infinity.

## References

1. __C. Ashbacher__, *The Pseudo-Smarandache Function and the Classical Functions of Number Theory*, **Smarandache Notions J.**, 9(1998), No. 1-2, 78-81.

2.__ K. Kashihara__, **Comments and Topics on Smarandache Notions and Problems**, Erhus Univ. Press, AZ., 1996.

3. __M. Bencze__, *OQ. 351*, **Octogon M.M.** 8(2000), No. 1, p. 275.

4.__ J. Sándor__, *On Certain New Inequalities and Limits for the Smarandache Function*, **Smarandache Notions J.**, 9(1998), No. 1-2, 63-69.

5. __J. Sándor__, *On the Difference of Alternate Compositions of Arithmetical Functions*, to appear.