We can call the count of primes in the permutations of the digits of n the prime 
diversity index of n. Then it follows that the limiting number (1 in this case) can 
be called the order of the diversity index. So this sequence is numbers with 
indices of prime diversity 1. Leading zeros have no effect in the results.
(PARI) 
/* Express the kth Permutation of the digits of n as a decimal*/
permute(n,k) =
{
local(ln,j,tmp,s=0);
n=Str(n);
ln=length(n);
n=eval(Vec(n));
for(j=2,ln,
tmp=n[k%j+1];
n[k%j+1] = n[j];
n[j]=tmp;
k = floor(k/j);
);
for(j=1,ln,s+=n[j]*10^(ln-j)); \\ Put it into decimal
s;
}
/* Count the number of distinct primes in the permutations of the
digits of n. Also the index of prime diversity of n.
*/
permprime(n) =
{
local(flag,m,x,j,k,y,ln,v,v2,v3,c=0,c2=0);
ln=length(Str(n))!;
v2=vector(ln+1);
v=vector(ln+1);
for(x=1,ln,
y=permute(n,x);
if(isprime(y),
c++;
v[c]=y;
);
);
v2=vecsort(v);
for(j=2,ln+1,
if(v2[j]<>v2[j-1],
c2++;
);
);
c2;
}
/* List the index of prime diversity of n where 
IPD is the numnber of permutations of order k.
*/
/* List the distinct primes in the permutations of the digits of n
where the count is greater than a constant value k. Call this
permutation primes of order k.
*/
listpermprime(n,k) =
{
local(x,y,ct,s);
for(x=1,n,y=permprime(x);if(y>k,
print1(x",");ct++;
);
);
ct;
}