Smarandache Near-to-Primorial Function

${\it SNTP}(n)$ is the smallest Prime such that $p\char93 -1$, $p\char93$, or $p\char93 +1$ is divisible by $n$, where $p\char93$ is the Primorial of $p$. Ashbacher (1996) shows that ${\it SNTP}(n)$ only exists

1. If there are no square or higher powers in the factorization of $n$, or

2. If there exists a Prime $q<p$ such that $n\vert(q\char93 \pm 1)$, where $p$ is the smallest power contained in the factorization of $n$.

Therefore, ${\it SNTP}(n)$ does not exist for the Squareful numbers $n=4$, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, ... (Sloane's A013929). The first few values of ${\it SNTP}(n)$, where defined, are 2, 2, 2, 3, 3, 3, 5, 7, ... (Sloane's A046026).

See also Primorial, Smarandache Function

References

Ashbacher, C. ``A Note on the Smarandache Near-To-Primordial Function.'' Smarandache Notions J. 7, 46-49, 1996.

Mudge, M. R. ``The Smarandache Near-To-Primorial Function.'' Abstracts of Papers Presented to the Amer. Math. Soc. 17, 585, 1996.

Sloane, N. J. A. A013929 and A046026 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.