Checking whether a kvalue makes a full covering set with algebraic factors not always very easy. The way I do it is to look for patterns in the factors of the various nvalues for specific kvalues. If there are algebraic factors, it's most common for them to be in a pattern of f*(f+2), i.e.:
11*13
179*181
etc.
In other cases there may be a consistent steady increase in the differences of their factors, which is especially tricky to find but indicates the existence of algebraic factors.
e.g. for the case R15 k=47
nvalue : factors
1 : 2^5 · 11
2 : 17 · 311
3 : 2^4 · 4957
4 : 31 · 38377
6 : 11 · 43 · 565919
8 : 199 · 1627 · 186019
10 : 17 · 61 · 13067776451
12 : 37 · 82406457849451
20 : 15061 · 236863181 · 2190492030407
Analysis:
For n=1 & 3 (and all odd n), all values are divisible by 2 so we only consider even n's.
For n=4, the two prime factors does not close.
For n=6 & 10, multiplying the 2 lower prime factors together does not come close to the higher prime factor so little chance of algebraic factors.
For n=12, the large lowest prime factor that bears no relation to the other prime factor means that there is unlikely to be a pattern to the occurrences of large prime factors so there must be a prime at some point.
R33 k=257:
nvalue : factors
1 : 5 · 53
2 : 2 · 4373
3 : 397 · 727
4 : 2^2 · 2381107
5 : 5^3 · 7 · 359207
7 : 11027 · 31040117
15 : 13337 · 706661 · 51076716238627
19 : 38231 · 14932493857679888742000509
For n=15 & 19 same explanation as R15 k=47
R36 k=1555:
nvalue : factors
1 : 11 · 727
2 : 31 · 37 · 251
3 : 67 · 154691
4 : 37 · 127 · 271 · 293
7 : 4943 · 3521755879
9 : 59 · 382386761790283
For n=7 & 9 same explanation as R15 k=47
The prime factors for n=12, n=15, and n=7 respectively make it clear to me that these kvalues should all yield primes at some point so you are correct to include them as remaining.
The highermath folks may be able to chime in and answer why there are an abnormally large # of k's that are perfect squares that end up remaining even though they don't have known algebraic factors for most bases. IMHO, it's because there ARE algebraic factors for a subset of the universe of nvalues on them but not for all of the nvalues. Hence they are frequently lower weight than the other k's but NOT zero weight and so should eventually yield a prime.
