@Axel Vogt Thanks, I'll look at the message boards.  I agree that Pari (and, for example, Sage) are tough for people like me, though. (i.e., people who do math but not so much computer science.)

Barry

@Markiyan Hirnyk  Markiyan, I'm wondering now, when you said that the coefficients make a big difference, did you mean, not that computing the coefficients takes up time, but that they make a big difference in some other way?

Barry

@Markiyan Hirnyk Markiyan, I see I've been misspelling your name, sorry for that too. i's and l's are hard for me to distinguish, but I should have been careful.

Barry

@Markiyan Hirnyk 

Marklyan, p.s., Lehmer's method is probably not as efficient as more recent ones.  A book edited by Edixhoven (Princeton U. Press, probably) "Computational Aspects of Modular Forms and Galois Representations" speaks to this question. Edixhoven apparently gives a method to compute τ(p) in time polynomial in log p (p prime.)  This is enough to compute the other values, but I don't know how quickly.

By the way, for a rather compressed but comprehensive intro to the Ramanujan function, see Serre's "A Course in Arithmetic" (Springer).

Barry

@Markiyan Hirnyk 

Marklyan, I tried to couch my last reply as tentatively as I could to indicate that it's just a guess.  (About the depth of the reason for the slowness of the Mathematica command, I mean.) I thought this was clear, but if my writing was inept in this respect, I'm sorry.

As far as the statement about Lehmer's method, (i.e. that the values of Ramanujan's tau function can be computed rapidly using Lehmer's method), it seemed rapid to me when I used it--much quicker than simply invoking Mathematica's tau-function command [not to be confused with Mathematica's Ramanujan-tau Dirichlet series command....].  

Here, I suppose I really am invoking my own experience re speedup without citing a source.  But you can test this yourself if you want.  Here is the reference to Lehmer's original paper describing his method:

Ramanujan's function τ(n), Duke Math. J., 10 (1943) 483-492.

A nice book-length survey of relevant material is "Ramanujan Revisited" (Academic Press).

Barry