Smarandache Ceil Function

A Smarandache-like function which is defined where $S_k(n)$ is defined as the smallest integer for which $n\vert{S_k(n)}^k$. The Smarandache $S_k(n)$ function can therefore be obtained by replacing any factors which are $k$th powers in $n$ by their $k$ roots. The functions $S_k(n)$ for $k=2$, 3, ..., 6 for values such that $S_k(n)\not=n$ are tabulated by Begay (1997).

$S_1(n)=n$, so the first few values of $S_1(n)$ are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ... (Sloane's A000027). The first few values of $S_2(n)$ are 1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, ... (Sloane's A019554) The first few values of $S_3(n)$ are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, ... (Sloane's A019555) The first few values of $S_4(n)$ are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, ... (Sloane's A007947).

See also Pseudosmarandache Function, Smarandache Function, Smarandache-Kurepa Function, Smarandache Near-to-Primorial Function, Smarandache Sequences, Smarandache-Wagstaff Function

References

Begay, A. ``Smarandache Ceil Functions.'' Bull. Pure Appl. Sci. 16E, 227-229, 1997.

``Functions in Number Theory.'' http://www.gallup.unm.edu/~smarandache/FUNCT1.TXT.

Sloane, N. J. A. Sequences A007947, A019554, A019555, and A000027/M0472 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Smarandache, F. Collected Papers, Vol. 2. Kishinev, Moldova: Kishinev University Press, 1997.

Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993.