OFFSET
1,1
COMMENTS
3-almost prime analog of A056809.
This sequence consists of the least of 3 consecutive 3-almost primes, or 4 or more consecutive 3-almost primes (i.e. n, n+1 and n+2 but not excluding n+3 also 3-almost prime). A067813 has some runs of up to 7 consecutive 3-almost primes (i.e. starting 211673). But there cannot be 8 consecutive 3-almost primes, as every run of 8 consecutive positive integers contains exactly one multiple of 8 = 2^3 and only 8 of all positive multiples of 8 is a 3-almost prime (i.e., all larger multiples have at least 4 prime factors, with multiplicity).
Primes counted with multiplicity. - Harvey P. Dale, Sep 04 2019
LINKS
D. W. Wilson, Table of n, a(n) for n = 1..10000
FORMULA
n, n+1 and n+2 are all elements of A014612.
EXAMPLE
a(1) = 170 because 170 = 2 * 5 * 17 and 171 = 3^2 * 19 and 172 = 2^2 * 43 are all 3-almost primes.
a(2) = 244 because 244 = 2^2 * 61 and 245 = 5 * 7^2 and 246 = 2 * 3 * 41 are all 3-almost primes.
a(3) = 284 because 284 = 2^2 * 71 and 285 = 3 * 5 * 19 and 286 = 2 * 11 * 13 are all 3-almost primes.
a(4) = 428 because 428 = 2^2 * 107 and 429 = 3 * 11 * 13 and 430 = 2 * 5 * 43 are all 3-almost primes.
a(5) = 434 because 434 = 2 * 7 * 31 and 435 = 3 * 5 * 29 and 436 = 2^2 * 109 are all 3-almost primes.
a(6) = 506 because 506 = 2 * 11 * 23 and 507 = 3 * 13^2 and 508 = 2^2 * 127 all 3-almost primes.
a(7), a(8), a(9) = 602, 603, 604 because of the record-setting 5 consecutive 3-almost primes: 602 = 2 * 7 * 43; 603 = 3^2 * 67; 604 = 2^2 * 151; 605 = 5 * 11^2; 606 = 2 * 3 * 101.
MATHEMATICA
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@2664, fQ@# && fQ[ # + 1] && fQ[ # + 2] &] (* Robert G. Wilson v, Jan 21 2006 *)
SequencePosition[Table[If[PrimeOmega[n]==3, 1, 0], {n, 3000}], {1, 1, 1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2019 *)
PROG
(PARI) is(n)=bigomega(n)==3 && bigomega(n+1)==3 && bigomega(n+2)==3 \\ Charles R Greathouse IV, Feb 05 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jan 21 2006
EXTENSIONS
Edited, corrected and extended by Robert G. Wilson v, Jan 21 2006
STATUS
approved