arithmetic progressions of numbers with the same prime signature

[David Wasserman]

Dear Seqfans,
    In this message I will use the abbreviation "APPS" for "arithmetic progression of numbers with the same prime signature".  I just came across some unedited sequences concerning APPS, and I've become very interested in the subject.   I'm trying to put together a definitive package of sequences concerning APPS, and I'd appreciate people pointing out relevant sequences already in the OEIS, and suggesting other sequences that should be computed.  Also, there's one sequence that I definitely need help with:

Let a(n) be the least number that begins an n-term APPS.  This sequence begins 1,2,3,5,5,7,7,11,11,11,11, as exhibited by the APPS
1
2, 3
3, 5, 7
5, 11, 17, 23
5, 11, 17, 23, 29
7, 37, 67, 97, 127, 157
7, 157, 307, 457, 607, 757, 907
11, 1210241, 2420471, 3630701, 4840931, 6051161, 7261391, 8471621
11, 32671181, 65342351, 98013521, 130684691, 163355861, 196027031, 228698201, 261369371
11, 224494631, 448989251, 673483871, 897978491, 1122473111, 1346967731, 1571462351, 1795956971, 2020451591
11, 1536160091, 3072320171, 4608480251, 6144640331, 7680800411, 9216960491, 10753120571, 12289280651, 13825440731
Notice that for 2 <= n <= 11, a(n) = nextprime(n).  I've verified that a(n) >= nextprime(n) for n up to 100, and I'm sure this holds for all n > 1, but I haven't figured out a proof.  I would conjecture that a(n) = nextprime(n) for all n > 1, but as n grows, these APPS will probably become very hard to find.  Is there a theorem or a well-known conjecture that implies they must exist?

Now for my listing of sequences:  
The sequences that got me started on this topic were
A087306(n) = least number that begins an n-term APPS with common difference n

and its companion 
A087307(n) =  least number that ends an n-term APPS with common difference n
(= A087306(n) + n(n - 1).)

I'm generalizing A087306 to a square array
T(n, d) = least number that begins an n-term APPS with common difference d:
        1    1    1   1  1  1    1   1 ...
        2    3    2   3  2  5   14   3
       33    3  155   3 77  5   51   3
    19940  213 7572 111     5 4214  69
   204323  213      201     5      402
380480345 1383      201  1333      402
          3091      481  1891      511
          8129           2159
(Apparently these APPS are easier to find for even d.)
Thus A087306 is the main diagonal.  A034173 is the first column.  The second column is probably worth submitting.  A086489 is the second row.  The 3rd row is worth submitting.  a(n) above is the smallest number in the nth row.

A087309(n) is the least number that ends an n-term APPS.  A086786 is a triangle made up of these APPS, A087308 gives their first terms, and A087310 gives their common differences.

A052213 lists all numbers that begin a 2-term APPS with common difference 1.  A052214 lists all numbers that begin a 3-term APPS with common difference 1.

That's all the APPS sequences I've found, except for a few others that are trivially related to these, such as A034174(n) = A034173(n) + n - 1.

A083788 concerns arithmetic progressions in which all members have distinct prime signatures.  I'll definitely include a cross-reference to that one, and maybe compute a few more sequences concerning such progressions.

Also, I'll probably want to repeat the whole exercise with "same number of divisors" instead of "same prime signature", and have cross-references between the corresponding sequences.

Please let me know if you submit anything along these lines.  I'm not planning to submit anything until I have the whole package assembled and cross-referenced to my satisfaction, and that could take several weeks.

Regards,
David