[seqfan] Smallest prime with a substring of exactly n zeros

[Felix Fröhlich]

Dear Sequence fans

a(n) is the smallest prime p containing a substring of exactly n zeros. The
sequence (with offset 1) starts

101, 1009, 10007, 100003, 1000003, 20000003, 100000007, 1000000007

Does the sequence always coincide with A037053? A counterexample would have
to be a prime having exactly n zeros in its decimal expansion but where
those zeros do not form a contiguous string. That prime would also need to
be the smallest prime with exactly n zeros. Does such a counterexample
exist?

Best regards
Felix Fröhlich