1,1

For n>=2 there cannot be more than 2^n - 1 consecutive n-almost primes. Is it known whether there always exists such a run of length 2^n - 1? If not, I conjecture so. This is confirmed to be true for terms through a(4). Terms here equal the last terms of corresponding finite sequences: a(3) = A067813(6). a(4) was computed by Don Reble as A067814(14). a(5) >= A067820(12).

a(4) is smaller than the number 488995430567765317569 found by Forbes. [From T. D. Noe (noe(AT)sspectra.com), Oct 29 2008]

Tony Forbes, Fifteen consecutive integers with exactly four prime factors, Math. Comp. 71 (2002), 449-452. [From T. D. Noe (noe(AT)sspectra.com), Oct 29 2008]

a(2) = 33 because 33, 34, 35 is the least run of three consecutive 2-almost primes (semiprimes).

Cf. A067813, A067814, A067820, A067821, A067822.

Sequence in context: A083459 A034173 A132519 this_sequence A003820 A112980 A109336

Adjacent sequences: A117966 A117967 A117968 this_sequence A117970 A117971 A117972

hard,nonn,new

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Apr 05 2006