But Dan wanted just the subset which yielded primes. As he implied, it's not even known that this isn't finite. For the l.c.m. of 1,2,...,n Hardy and Wright use U(x) and one can no doubt find other notations in number theory texts. A clumsy but well accepted one is [1,2,...,n]. The important thing is that it's \psi(x) e where \psi(x) is pretty generally accepted in the prime number theory world. R. On Wed, 14 Apr 2004, Richard Schroeppel wrote:

...

2. Is there a nice asymptotic expression for the number of E_n < x ???

The product of the primes <= X is (very roughly) e^X. There are about X/logX primes <= X, and their "average" magnitude is (very roughly) X.

Suggested notation: We are using X# for this product. I suggest extending this to X## for the LCM of numbers <= X. This is the same as the product (for each prime) of the largest prime power <=X.

So 6# = 2.3.5 = 30, while 6## = 4.3.5 = 60.

The logs of X# and X## are standard functions in analytic number theory.

Rich rcs@cs.arizona.edu

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