Previous Prime

The previous prime function  gives the largest prime less than . The function can be given explicitly as

where  is the th prime and  is the prime counting function. For , 4, ... the values are 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, ... (OEIS A007917).

The previous prime function was implemented in versions of the Wolfram Language prior to 6 as PreviousPrime[n] (after loading the package NumberTheory`NumberTheoryFunctions).

Finding the previous prime before  gives the largest -digit prime. For , 2, ..., the first few of these are 7, 97, 997, 9973, 99991, 999983, 9999991, 99999989, ... (OEIS A003618). The amounts by which these are less than  are 3, 3, 3, 3, 27, 9, 17, 9, 11, 63, ... (OEIS A033873).

REFERENCES:

O'Hara, F. "More Prime Trivia." J. Recr. Math. 20, No. 3, 209-210, 1988.

O'Hara, F. "The Smallest and Largest -Digit Primes for ." J. Recr. Math. 22, No. 4, 277-279, 1990.

Ondrejka, R. "Prime Trivia." J. Recr. Math. 14, No. 4, 285, 1981-1982.

Sloane, N. J. A. Sequences A003618/M4452, A007917, and A033874 in "The On-Line Encyclopedia of Integer Sequences."