Counting Carmichael numbers with small seeds

Let As be the product of the first s primes, let Ps be the set of primes p for which p−1 divides As but p does not divide As, and let Cs be the set of Carmichael numbers n such that n is composed entirely of the primes in Ps and such that As divides n − 1. Erdős argued that, for any ε > 0 and all sufficiently large x (depending on the choice of ε), the set Cs contains more than x1−ε Carmichael numbers ≤ x, where s is the largest number such that the sth prime is less than lnxε/4. Based on Erdős’s original heuristic, though with certain modification, Alford, Granville, and Pomerance proved that there are more than x2/7 Carmichael numbers up to x, once x is sufficiently large. The main purpose of this paper is to give numerical evidence to support the following conjecture which shows that |Cs| grows rapidly on s: |Cs| = 22 s(1−ε) with lims→∞ ε = 0, or, equivalently, |Cs| = A2 s(1−ε′) s with lims→∞ ε ′ = 0. We describe a procedure to compute exact values of |Cs| for small s. In particular, we find that |C9| = 8, 281, 366, 855, 879, 527 with ε = 0.36393 . . . and that |C10| = 21, 823, 464, 288, 660, 480, 291, 170, 614, 377, 509, 316 with ε = 0.31662 . . .. The entire calculation for computing |Cs| for s ≤ 10 took about 1,500 hours on a PC Pentium Dual E2180/2.0GHz with 1.99 GB memory and 36 GB disk space.