nt.number theory - The missing Euler Idoneal numbers

The reference

Weinberger, P. J.: Exponents of the class groups of complex quadratic fields, Acta
Arith. 22 (1973), 117–124.

was what Matthias Schütt and I cited for the fact that "there is at most one further imaginary quadratic field with class group exponent 2". (The context is constructing at least one K3 surface for each of the 65 [known] idoneal numbers; see http://arxiv.org/pdf/0809.0830.)

For the specific question about 3 or more primes, Pete Clark's argument reduces this to the solution of the class number $h$ problem for $h=1,2,4$. Mark Watkins' paper

Watkins, M.: Class numbers of imaginary quadratic fields, Math. of Comp. 73 (2003) #246, 907-938

which does all $h leq 100$, has been mentioned already. He starts by reviewing earlier work, ending with "Arno's thesis [2] and subsequent work with
Robinson and Wheeler [3] and the work of Wagner [44], which together complete
the classication for all $N leq 7$ and odd $N leq 23$." On page 5 of Watkins' paper (immediately following the statement of Lemma 3) he notes that "From Gauss's theory of
genera [10], [8], we know that $2^{omega(d)-1}$ divides $h(-d)$ where $omega(d)$ is the number of distinct prime factors of d...", which corresponds to Pete's observation.