1,1

Using a musical analogy, each run is a "crescendo" of primality where each subsequent member of the run is gradually "more prime" in the sense of having one fewer distinct prime factors (see A001221). These a(n) are the peaks of crescendos of increasing length. a(7) is greater than 60000000.

This sequence and A072663 were inspired by A068069, where the members of the runs have n different numbers of distinct prime factors, 1 through n, but where the order is not specified.

a(1)=2 because 2 is prime and therefore the smallest integer with exactly one distinct prime factor. a(2)=7 because 6=2*3 and 7 (prime) is the smallest run of consecutive integers with exactly 2 and 1 distinct prime factors in that order. a(3)=107 because 105=3*5*7, 106=2*53 and 107 (prime) is the smallest run with exactly 3, 2 and 1 distinct prime factors in that order. Note that a(1), a(2), a(3), a(5) and a(6) are prime but that a(4)=2187=3^7 is not.

Cf. A072663 (smallest start with run pattern 1, 2, ..., n), A072665 (center with run pattern n+1, n, ..., 2, 1, 2, ..., n, n+1), A068069 (run order not specified), A001221 (omega(n)).

Sequence in context: A102747 A122524 A162634 this_sequence A045310 A000157 A034902

Adjacent sequences: A072661 A072662 A072663 this_sequence A072665 A072666 A072667

nonn

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 30 2002

a(7) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Jan 24 2009