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%I #27 Nov 24 2020 03:11:10
%S 170,244,284,428,434,506,602,603,604,637,962,1074,1083,1084,1130,1244,
%T 1309,1412,1434,1490,1532,1556,1586,1604,1634,1675,1771,1885,1946,
%U 2012,2013,2035,2084,2091,2092,2162,2396,2404,2522,2523,2524,2525,2634,2635
%N Numbers n such that n, n+1 and n+2 are products of exactly 3 primes.
%C 3-almost prime analog of A056809.
%C 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).
%C Primes counted with multiplicity. - _Harvey P. Dale_, Sep 04 2019
%H D. W. Wilson, <a href="/A113789/b113789.txt">Table of n, a(n) for n = 1..10000</a>
%F n, n+1 and n+2 are all elements of A014612.
%e a(1) = 170 because 170 = 2 * 5 * 17 and 171 = 3^2 * 19 and 172 = 2^2 * 43 are all 3-almost primes.
%e a(2) = 244 because 244 = 2^2 * 61 and 245 = 5 * 7^2 and 246 = 2 * 3 * 41 are all 3-almost primes.
%e a(3) = 284 because 284 = 2^2 * 71 and 285 = 3 * 5 * 19 and 286 = 2 * 11 * 13 are all 3-almost primes.
%e a(4) = 428 because 428 = 2^2 * 107 and 429 = 3 * 11 * 13 and 430 = 2 * 5 * 43 are all 3-almost primes.
%e a(5) = 434 because 434 = 2 * 7 * 31 and 435 = 3 * 5 * 29 and 436 = 2^2 * 109 are all 3-almost primes.
%e a(6) = 506 because 506 = 2 * 11 * 23 and 507 = 3 * 13^2 and 508 = 2^2 * 127 all 3-almost primes.
%e 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.
%t fQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@2664, fQ@# && fQ[ # + 1] && fQ[ # + 2] &] (* _Robert G. Wilson v_, Jan 21 2006 *)
%t 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 *)
%o (PARI) is(n)=bigomega(n)==3 && bigomega(n+1)==3 && bigomega(n+2)==3 \\ _Charles R Greathouse IV_, Feb 05 2017
%Y Cf. A014612, A056809, A067813.
%Y Subsequence of A180117.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Jan 21 2006
%E Edited, corrected and extended by _Robert G. Wilson v_, Jan 21 2006
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