Smarandache Near-to-Primorial Function

 is the smallest prime such that , , or  is divisible by , where  is the primorial of . Ashbacher (1996) shows that  only exists

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

2. If there exists a prime  such that , where  is the smallest power contained in the factorization of .

Therefore,  does not exist for the squareful numbers , 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, ... (OEIS A013929). The first few values of , where defined, are 2, 2, 2, 3, 3, 3, 5, 7, ... (OEIS A046026).

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. Sequences A013929 and A046026 in "The On-Line Encyclopedia of Integer Sequences."