A072664 End of smallest run of n consecutive integers with n, n-1, ..., 1 distinct prime factors in that order.

2, 7, 107, 2187, 578309, 12239309, 45640671719, 29591195200067

Offset

1,1

Comments

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 was 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.

Links

Table of n, a(n) for n=1..8.

Example

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.

Crossrefs

Cf. A086560 (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: A307329 A162634 A235470 * A352046 A045310 A224445

Adjacent sequences: A072661 A072662 A072663 * A072665 A072666 A072667

Keyword

nonn

Author

Rick L. Shepherd, Jun 30 2002

Extensions

a(7) from Donovan Johnson, Jan 24 2009

a(8) from Donovan Johnson, Jul 19 2011

Status

approved