Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).
The prime factors of 10^m + 1 are also prime factors of 10^km + 1, where k is odd, so a(n) and a(km) have those prime factors in common.
Expanding on the comment in A083359 regarding finding large terms, we can generate large doublet terms in the following way:
-- for any composite number M = 10^m + 1 try to compose a prime number p of length m from the prime factors of M. Generally this will require the factors to overlap to reduce the length to m.
-- the factors can wrap around to the beginning of p. For example, if M has a factor of 137 then p can be of the form 7...13.
-- the term is formed by concatenating p with itself to form a(n) = p||p. The resulting number will consist entirely of the concatenation of its prime factors with allowed overlap as in A083359.