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Borrowing from musical terminology, these could be considered "swells" of primality - first a crescendo ("more prime"), then a decrescendo ("less prime"). a(43), if it exists, is greater than 70750000. The corresponding sequence but counting prime factors with multiplicity (A01222A001222) has only two terms (2, 5) cannot exist because either the number immediately before or after the any odd center would be a multiple of 4, > 5 equals 4k for some k >= 2, and thus would always have to be 4has at least three prime factors, not exactly two, when duplicates are counted.
a(0) = 2 (prime) is the smallest number with one prime factor. a(1) = 11 as 10 (=2*5), 11 (prime) and 12 (=2^2*3) have 2,1,2 distinct prime factors (A01221A001221), respectively and there is no smaller center of such a run. a(2) = 2917 as 2915 (=5*11*53), 2916 (=2^2*3^6), 2917 (prime), 2918 (=2*1459) and 2919 (=3*7*139) have 3,2,1,2,3 distinct prime factors and there is no smaller such run.
Comment expanded and small typos fixed by Rick L. Shepherd, Jun 22 2017
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a(3) > 10^63. - Hiroaki Yamanouchi, Sep 25 2014
hard,nonn,more,bref
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