A072665 Center of smallest run of 2n+1 consecutive numbers with exactly n+1,n,...,2,1,2,...,n,n+1 distinct prime factors, respectively.

2, 11, 2917

Offset: 0

Author

Rick L. Shepherd, Jul 30 2002

Keywords

• hard
• nonn
• more
• bref

Comments

Borrowing from musical terminology, these could be considered "swells" of primality - first a crescendo ("more prime"), then a decrescendo ("less prime"). a(3), if it exists, is greater than 70750000. The corresponding sequence but counting prime factors with multiplicity (A001222) has only two terms (2, 5) because either the number immediately before or after any odd center > 5 equals 4k for some k >= 2, and thus has at least three prime factors, not exactly two, when duplicates are counted.

a(3) > 10^63. - Hiroaki Yamanouchi, Sep 25 2014

Examples

			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 (A001221), 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.
		

Crossrefs

Cf. A072664 (smallest finish with run pattern n, ..., 2, 1), A086560 (smallest start with run pattern 1, 2, ..., n), A001221 (omega).

Extensions

Comment expanded and small typos fixed by Rick L. Shepherd, Jun 22 2017